Every essay
What can move
Before a mechanism does anything it must be able to. Mobility is countable, and the count can be wrong — which is how some of the most useful mechanisms ever built were nearly ruled out.
What decides whether it moves
Before a mechanism does anything it has to be able to. Two bars pinned to a point cannot move; three can. The count that separates them is one subtraction, it is the first thing anybody computes about a machine, and it can be wrong in a way that no amount of care with the arithmetic will catch.
Counting and measuring mobility
Grübler's criterion counts links and joints and never asks how long anything is. The rank of the constraint Jacobian measures the lengths and never asks what a joint is. Two calculations with no inputs in common, producing one number — which is the only arrangement under which agreement is evidence.
The mechanism Grübler says cannot move
Three parallel bars between two frames. Five links, six pins, and the criterion every engineering course teaches gives zero degrees of freedom — a structure. It is a mechanism, it is in drafting machines and locomotive coupling rods, and the formula cannot see why.
What each joint takes away
Grübler's formula has a 2 in it for pins and a 1 for cam contacts, and those numbers are not conventions to be memorised. They are the number of constraints each kind of joint imposes, they are measurable as the rank of a matrix, and miscounting one of them is the commonest way the formula is got wrong.
The formula is repaired by the thing it replaced
Kutzbach's count is wrong about most of the mechanisms worth building, and every textbook gives the same repair — add back the constraints that were imposed twice. The repair works on every loop in these essays. It is also not a formula, because the number it adds cannot be read off the joint graph.
The count that counts the wrong thing
Mobility has meant one number for six fields, because until a wheel appeared no mechanism could tell two questions apart. A rolling wheel has two velocity freedoms and a three-dimensional reachable set, and the formula that gives 2 is not wrong — it is answering the question about instants when the question anybody asks is about intervals.
The count was right and the name was wrong
The constraint field has checked Grübler's count against a Jacobian rank since the foundation, and the two disagree only where the geometry is special. Here is an assembly where they agree, where both are correct, and where the mechanism does not have the number of links it is described as having.
The freedom that is a set
Grübler's rule has been in this collection since its first essay, and it adds up numbers. Each of those numbers is the dimension of a group of displacements, and the group has eleven siblings the number cannot distinguish. The count is not wrong; it is a projection, and this is what the projection discards.
One freedom and four hundred links
Braced, the machine compiled from a quintic has 1,249 equations in 1,096 unknowns and a Grübler count of minus a hundred and fifty-three. It turns. The rank of its constraint Jacobian is 1,095, so its mobility is one — and every one of the hundred and fifty-four surplus equations was added deliberately.
One freedom, and a motion that never repeats
Mesh a gear on each crank of a five-bar and the count and the rank agree that one freedom is left, at every gear ratio. Whether the machine ever comes back to where it started is a different question, and neither instrument can see it: at 3 to 2 it is home after two turns, at 37 to 23 after twenty-three, and at the golden ratio never, with its nearest returns at the Fibonacci numbers.
A parallelogram a micron wrong
A parallelogram linkage sits exactly where two kinds of four-bar meet, so a parallelogram that has actually been made is always one of four other machines. Make one bar a micron wrong on a 300 mm frame and the input stops a tenth of a degree short of lying flat, or the output turns round there with an acceleration that grows as one over the square root of the error. A third bar turns the square root back into a misfit of one micron.
Nine bars that ought to be rigid
Join each of three joints to each of three others and the nine bars leave no freedom, by the count and by the rank, wherever the joints are. Put one set on a line and the other on a line at right angles and the framework moves, all the way round a loop, with no bar repeating any other: take away any one of the nine and the motion is unchanged, take away any two and it gains a freedom. Tilt the lines by a degree and it still has a freedom by rank and cannot move at all.
A length error is undone by its own size
A parallelogram built a thousandth wrong loses its change point, and the two motions it could have chosen between end up a tenth of a radian apart — the square root of the error rather than the error. The radial play that joins them again is a thousandth exactly: not of that order, that number. It is the same number a third crank charges the same linkage in misfit, and no pin is worth more of it than any other.
The right angle as a tolerance
Dixon's nine bars move only when their two lines are exactly perpendicular, and a framework built a degree off square has a freedom by rank and no motion at all. Give its joints clearance and it moves a bounded distance: the play each bar needs is proportional to the tilt and to the square of the travel, one constant serves every tilt, and all nine bars end up at that play exactly. Then it reaches its first crossing, where the push is as large as the motion ever makes it, and a hundredth of a push further costs fifty-eight times the law — square or not.
Every change point lies flat
A parallelogram a thousandth wrong is rejoined by a pin clearance of exactly a thousandth, at any of its four bearings. The obvious guess is that an ordinary change point — a linkage on one Grashof boundary with no equal bars — would need a clearance with a constant in front and would reveal which bearing is loose. It does neither, because every change point has its four joints on one line. What does acquire a constant is the angle: 2√(2bδ/c(g + a)) when the circuits separate, and a stall constant with no coupler or output in it at all.
The clearance that makes the framework generic
A clearance lets Dixon's nine bars move, and it also moves their joints off the two lines that let them move at all. Six joints off a conic make the framework rigid by rank, with a smallest singular value proportional to its distance from a conic. The same clearance that did the damage pays for it fourteen times over: a framework built anywhere inside its clearance discs is pushed to its first crossing using at most 7% of the allowance those discs give. The tilt of the lines decides how far it goes; its own joint errors change that by about 1%.
The price of the other family
At its crossing, Dixon's framework passes from one family of placements to another, and past it every push names two placements, one in each. A framework built with its joints anywhere inside a clearance ρ has no motion to follow, so its play has to carry it. The price of going on is proportional to ρ, from 0.002ρ to 0.20ρ across ten frameworks, exactly as a parallelogram's change point is — and reversing every joint error leaves it unchanged. Eight of the ten find the far family dearer than their own and two find it cheaper, and all of them find it far inside the 2ρ their clearance gives.
The price is one self-stress
A Dixon framework built with its joints a little out of place has to be carried past its crossing by play, and the play it needs was measured by bisection, framework by framework. It has a closed form. Pinned and pushed, the framework carries exactly one self-stress on each family, and the least play is the errors' bar misfits weighted by that stress, over the stress's size. The formula matches every measured price to 0.6%, says why two frameworks of ten prefer the far family, and finds that one in twelve does.
A push turns back at the crossing
Dixon's framework as built has play enough to go on past its first crossing onto either family, and the question left open was which one a push on B₂ chooses: the family whose self-stress is cheaper, or something else. Neither. Dixon's own formula says that near the crossing every joint but W₂ moves as the square of the distance along the motion, so the push is at a fold there on both sides, and no push can carry the framework through. Ten frameworks as built, pushed to their turning point and pulled back, all retrace, including the two that are cheaper beyond. Driven instead by W₂'s side of the black line, all ten go through, needing exactly the worst price on their path, and the price is continuous through the crossing.
A drive that goes round is an angle
Dixon's framework moves round a loop through four crossings, and a push on B₂ folds at two of them while a drive on W₂'s side folds at the other two. A lever moving both was proposed as the drive that would go all the way round. It cannot, and no drive that is one number read off the joints can: round the whole loop B₂ and W₂ sit at √2 cos ψ and √2 sin ψ on their two lines, the ends of a trammel bar, so the loop is a circle and every such number has a top and a bottom on it. A lever only moves its two folds off the crossings. What goes round is an angle. Four of the framework's eighteen joint angles turn once as it goes once round, never stopping, and a motor at W₂ takes ten frameworks as built all the way round at the worst price on the loop — 1.00 to 1.46 times their worst crossing — after which every turn repeats the first.
Put the motor where the rocking bar is longest
Dixon's framework goes round its loop with its two nearest joints on the ends of a trammel bar, and a motor at a joint angle with that bar for a side carries it round with no fold. With the nearest joints starting at different distances it looked as though the loop would become an ellipse and some angle would nearly stop. The loop stays a circle at every ratio, because the bar between the two joints is a bar. Each of the four turning angles runs at one radian per unit of the loop, plus or minus the rate of its other side, which rocks fastest at a crossing and there turns at the trammel bar's length over its own. That gives each angle's speed ratio in closed form, says none of them ever stops, and puts the motor at whichever joint has the longest rocking bar.
Linkages
Four bars and four pins is the smallest interesting machine there is. Everything about which link turns, how hard it pushes and where it jams follows from the four lengths.
Four bars and four pins
The smallest interesting machine there is. Four lengths decide everything about it — which link can turn all the way round, how hard it pushes, where it stops and whether it can be assembled at all — and every one of those is a number that falls out of a solve rather than a judgement about a drawing.
Grashof, predicted and then swept
Add the shortest link to the longest. If the total does not exceed the other two, some link can turn a full revolution. It is a sentence about four numbers, it was published in 1883, and it is the kind of claim these essays refuse to print without measuring — so every linkage here is also asked for all 360 positions and required to agree.
The transmission angle
The angle at which the coupler meets the rocker decides how much of an applied force becomes useful output torque and how much goes into the bearings. It is pure geometry, it is computed here from every solved position rather than from a formula, and it is the number a linkage is judged by after Grashof has said it turns.
The slider-crank
Replace one pin of a four-bar with a slide and you get the mechanism in every reciprocating engine ever built. Its stroke is exactly twice the crank throw and does not depend on the connecting rod at all. Everything else about the motion depends on the rod, including the part that is always described as a sine wave and is not.
The return stroke is quicker
A crank-rocker's output stops at two definite places, and the crank angles at which it does are calculable without touching a solver. The interesting number is not where they are but how far apart — because the crank turns at a constant speed and the output covers the same swing twice in unequal times.
One chain, four mechanisms
Which link of a four-bar is bolted to the bench is not a property of the chain. It is a decision about where the bench is, and making a different one gives a mechanism that looks and behaves completely differently while being, as a chain, the same object — which is why the Whitworth quick-return and the oscillating-cylinder engine are both a slider-crank.
Six bars, and what the extra dyad buys
Every linkage so far has had four bars, because four is the smallest closed chain that moves. The next one up is six, not five, and six is where the subject stops being one family and becomes a taxonomy — two chains, distinguished entirely by whether the two links carrying three joints happen to touch.
A dwell made from a curve
Parts of a coupler curve are very nearly circular arcs. Put a link of the arc's own radius on the coupler point and its far end stands almost still while the point runs along it, so the output dwells — 146° of crank inside a one-degree band, against 42° for the four-bar it is built on. A dwell linkage does not stop. It moves less than the tolerance, and how much less is a number.
Grashof is a shape test
The oldest classification in the subject compares sums of lengths, so it is unchanged by making the machine bigger — which means a protractor recovers it exactly without recovering a single length. What it does not recover is the margin, and the margin is what says whether the classification is safe.
Eight kinds of four-bar
Grashof's condition gives a four-bar one of four names and calls every linkage that fails it a triple rocker. Three signed sums of the lengths give eight, and a census of four thousand random linkages finds every one moving exactly as its signs say — because the planes where those sums vanish are the only places a four-bar's motion can change its kind.
A swing and a time ratio
A shaper's specification gives the rocker's swing and how much quicker the return must be than the cut, and those two numbers do not fix a linkage. They leave a one-parameter family of crank-rockers on the arcs of two circles, every member exactly right, and the transmission angle chooses between them — which is also what decides that a 60° swing cannot return more than 1.207 times as fast and keep 40°.
Four kinds of slider-crank
An offset slider-crank is a four-bar whose output bar and ground have grown without bound, and in that limit the three signed sums that sort four-bars into eight kinds lose one of their signs. Four kinds survive. A census of four thousand finds every slider-crank moving as its region predicts, the textbook condition for a full crank turn turns out to be one region exactly, and each of the four kinds that vanish is carried, at a length that can be written down, into the survivor that shares its other two signs.
A drag link ahead of a crank-rocker
A crank-rocker with a 60° swing keeps a transmission angle of 40° only up to a time ratio of 1.207, and at a ratio of 2 no crank-rocker keeps even 20°. Drive its crank from the output of a drag link, whose cranks both turn but not at the same speed, and a pair in which each stage keeps 40° returns 2.71 times as fast as it works. The phase between the two stages decides almost all of it: the same two linkages give anything from 1.003 to 2.71.
Three rotations, and four benches
The slider-crank chain's four inversions are four famous machines, and they are not four classifications. Four links make six pairs, one of those pairs cannot rotate at all because a slide is a rotation of nought, and the five that are left are three quantities between them. So an inversion chooses which two of the chain's three rotations sit at its bench, and the four regions of length space already say what all three do.
A quick return that cuts evenly
A drag link ahead of a crank-rocker buys a shaper its time ratio of 2 with both stages at 40°. The drag link that buys the most ratio drives the cut unevenly — its fastest speed through the middle of the stroke is 2.77 times its slowest — and a different drag link at a different phase reaches 2.03 with a ratio of 1.21, which is more even than the crank-rocker driven alone at constant speed. Up to a ratio of about 2.4 the second stage can improve both specifications at once.
Fifteen pairs and one cut
A six-bar has fifteen pairs of links, and each pair has a relative rotation. Followed round every circuit of more than four thousand random Watt and Stephenson six-bars, those fifteen rotations come from six whole-number windings. Almost always they take two values, so the links split into a group that turns once against the rest. How many inversions can be driven is then the number of links touching that cut: nought, three, four or five, never one, two or six. One chain in ten can have different cuts on different circuits, and a Stephenson chain can turn its rocker twice against its dyad.
The first stage does not know what it drives
A drag link that evens a shaper's rocker evens a ram driven by an offset slider-crank just as well — within 3.8% of the best that ram can have — once it is turned by the angle between the two machines' stroke starts, to within a degree. The two outputs cover their working strokes alike to 1.5% when they share an own time ratio, so what the first stage has to know about the second is that ratio and where its stroke begins. The exception is a centred slide with a short rod, whose skewed stroke moves the best phase by 12° more than the rule says and makes the train more even than any rocker's.
A centred slide has a best rod, and an offset beats it
Behind a drag link that brings a shaper's ram to a time ratio of 2, a centred slide's rod is a knob with a best setting. Shorten it from twenty cranks and the most even cut improves from 1.435 to 1.246 at a rod of two cranks, then worsens to 1.401 before the transmission angle reaches 40°, because the rod's skew moves the ram's speed peak where a drag link can phase against it only until the peak stops moving and just grows. An offset beats it: at every worst transmission angle up to 65° some offset slide is more even than any centred one, the best at 1.181, and on the most offset slide the rod hardly matters.
A centred ram cannot be timed past its drag link
A shaper's drag link and slider-crank were held to a combined time ratio of 2, and the centred slide's rod turned out to have a best length. Asking for a quicker return was supposed to shorten that rod and narrow the offset slide's lead. The first happens; the second runs the other way. A centred slide's dead centres are half a turn of its crank apart whatever its rod, so its ratio is the drag link's split of half a turn and nothing else: 2.304 on the grid, 2.443 off it, and no rod moves it. Below that ceiling the centred slide's best cut worsens from 1.118 to 1.466 and the offset slide's lead grows from 1% to 28%. Above it only an offset slide with a short rod reaches, and no slide keeping 45° reaches 3.
A ratio of three costs forty degrees
A shaper's two-stage drive, a drag link ahead of a slider-crank, could not be given a return three times as quick as its cut while its drag link kept 40° and its slide 45°. Drawn as a frontier, the largest ratio against the worst transmission angle both stages keep, a ratio of 3 costs 40.0° in both. The two stages in series reach 95.7% to 99.0% of the product of their separate ceilings at every angle from 30° to 60°, so two stages relax one stage's bound very nearly by a product. Shared unequally, every degree the drag link gives up buys the slide between one and a half and three: the drag link is the stage to let lean. And the trains on the frontier cut badly, the ram's fastest speed in the middle of its cut 3.2 times its slowest at a ratio of 3. On the way, the drag link's own ceiling at 40° turns out to be 2.443, not the 2.365 a single search had found.
Five degrees buy an even cut at a ratio of three
A shaper's two-stage drive reaches a quick return of 3 only by keeping no more than 40° of transmission angle in both stages, and the one train that does cuts unevenly, its ram 3.25 times faster at one point of the cut than at another. Searched for the most even cut that still returns three times as fast, the trains at 40° do little better, 2.97. But the price falls steeply below it: 1.90 at 37°, 1.42 at 35°, 1.22 at 30°. By 35° a ratio-3 shaper cuts as evenly as the best ratio-2 trains did. The even trains leave the long, nearly folded drag links that buy the most ratio for compact ones, and not one of them takes a longer rod than its angle requires.
The paths points trace
A point on a coupler draws a curve of degree six. Choosing the linkage that draws the curve you want is the oldest hard problem in the subject, and straight lines were the hardest of all.
What a coupler point draws
A point rigidly attached to the coupler of a four-bar traces a curve of degree six. Move the attachment a little and the curve changes a great deal. For most of the twentieth century the practical way to find the linkage that draws a wanted curve was to look it up in a book of printed atlases.
The straight-line problem
Before 1800 a long true flat surface was harder to make than almost anything else, so guiding a piston straight without a slide was worth solving. Watt's answer was an approximation. Measuring how good an approximation, over how much of the stroke, turns out to be a more interesting question than whether it is exact.
Peaucellier and the exact answer
Eighty years after Watt settled for an approximation, a French army officer found a linkage that draws an exactly straight line from pin joints alone. It works by inversion in a circle, the product it holds constant is measurable, and solved here it comes out straight to 10⁻¹⁶ of its span.
Where the coupler is turning
At every instant the coupler of a four-bar is rotating about a single point — not a pin, and usually not on the mechanism at all. Track that point in two different frames and you get two curves which, rolled on each other without slipping, reproduce the coupler's motion exactly. The bars are one way of producing it and not the motion itself.
Three linkages, one equation
Roberts's theorem says three different four-bars draw the same coupler curve. Fitted separately for the sextic that vanishes on each trace, on a common normalisation, the twenty-eight coefficients agree across all three to 5 × 10⁻⁷ — a test of the theorem that shares nothing with the construction the cognates came from.
The curve the other assembly draws
A crank-rocker's two assemblies do not share a coupler curve. Each draws a whole closed oval of its own through a full turn of the crank, the two ovals never meet, and both are the zero set of one sextic, so the equation a machine's own motion determines also describes a second machine it can never become.
A point the machine never reaches
Every coupler curve has three finite double points, and an odd number of them are real, so no coupler curve has none. On the standard crank-rocker the only real one is a point of the curve that neither assembly ever visits, that no contour plot can find, and that sits on the circle through the three pivots of Roberts's cognates.
A sextic that comes apart
A parallelogram chain's coupler curve is not one curve. Its sextic divides exactly by a circle, leaving a quartic, and each factor is one of the two things the machine can do. The division leaves rounding and nothing else, a coupler one millionth too long leaves a remainder a million times larger than rounding, and the two factors meet at the two places where the machine has to choose.
A symmetric curve from a lopsided machine
A four-bar with a crank of 1, a ground of 3 and a rocker of 2.5 has no symmetry anywhere in it. Make the rocker, the coupler and the arm from the rocker pin to the tracing point one length, and the curve it draws is its own mirror image to 4 × 10⁻¹⁵, about a line through the rocker pivot turned from the ground line by exactly half the coupler's angle at that pin.
The area a coupler point encloses
Trace a point on the line through a crank-rocker's two moving pins and the region its curve encloses has area (1 − u)πa²: the crank pin's own circle, scaled by how far along the line the point sits. No ground, coupler or rocker length appears in it. Four machines that share only a crank enclose 2.199115 each, to 3 × 10⁻¹⁴.
Where three machines keep one area
Roberts's theorem gives every four-bar two others that draw the same coupler curve, and so enclose the same areas. Measured, they do, to 10⁻¹² — but each keeps the area in a different place: the crank-rocker in its crank pin's circle, the double rocker in its coupler's turn, the third machine in its output pin's circle. What is left over has no closed form, and it is one number in all three.
The flattest dwell is not the longest
A coupler curve that is its own mirror image has no odd terms in its distance from a circle centred on the mirror line, so one angle of the coupler can remove the fourth-order term and leave a dwell of sixth order, with no search of the curve. A six-bar built there dwells for 58.7° of crank inside 0.1% of its swing. Turned two degrees away from that angle it dwells for 80.1°, and the searched six-bar for 24.4°.
A null space of fifteen is not noise
Points traced on one motion of a parallelogram four-bar leave a degree-six fit with fifteen polynomials that vanish on them, behind a drop of thirteen decades: the circle times every quartic. Half an oval of an ordinary coupler curve leaves two to five, behind drops of two. The count is the same kind of number in both cases, and only the drop beside it says which one is algebra.
The kind is decided before the lengths are
Roberts's construction hands a four-bar two others that draw its curve, and which of the eight kinds those two are is settled by the kind of the first — not by its lengths within that kind, and not by where the tracing point sits. Twenty-four thousand chains at five tracing points produce no exception, and the reason is one line: the tracing point enters the construction only as a scale, and a region is scale-blind.
The dip that buys the dwell
A symmetric coupler curve's six-bar dwells longest a little below the angle that makes its error sixth order, and three bands gave three numbers. Across three and a half decades of tolerance the best angle's distance from that angle grows as the cube root of the band, both dwells as its sixth root, and their ratio is (27/4)^(1/6) = 1.3747 at every band — because the best machine spends the whole band on one dip and ends its dwell where the output comes back.
A closed form belongs to a rotation
The three four-bars that draw one coupler curve carry the closed-form part of its area in different terms of the formula. Substituting Roberts's construction into those terms shows why: every term of a cognate is a term of the original, moved to another column. Over 480 random chains in all eight regions the three machines' closed forms agree as multisets to 10⁻¹⁵ and disagree column by column. Each closed form belongs to one of three rotations, and each machine carries each rotation in exactly one of its members.
A slide exchanges the crank and the rod
Roberts's construction gives a four-bar two cognates. Grow the rocker into a slide and one of them leaves with it; the other becomes a slider-crank whose crank is the original's rod and whose rod is the original's crank, the whole machine turned and scaled by the tracing point. So a slide keeps two of the figure's three rotations and freezes the third, the closed forms of the area move between crank and rod, and every slider-crank whose crank turns has a cognate whose crank cannot.
A second slide keeps only the rod
A slide turned a four-bar's rocker into a line and left one cognate, the slider-crank with its crank and rod exchanged. Turn the crank into a slide as well and the exchange stops producing a machine, since λ times a sliding pin's position is a vector that shrinks to nothing twice a turn. The cognates do not vanish; they multiply. Every double slider's ellipse is drawn by any two points of either of two rolling circles — a two-parameter family on each — and the one rotation left, the rod's, is carried by every one of them, with the original on one circle and against it on the other.
A third slide moves only through the crossing
Every point of a double slider's rolling circle runs on a line through the slides' crossing, so a rod can ride a third block on a third slide there, and a fourth and a fifth. The count gives nought, then minus one, then minus two; the rod turns freely every time, because each slide past the second is implied by the first two. Build the third slide slightly wrong and the count wins, in a way that depends on the error's kind: moved parallel, the rod turns freely inside the block's clearance and does not assemble outside it; turned by ε about the crossing, it moves only in two windows a turn, of 2·arcsin(c/(2r sin ε)) each, where the third pin passes through the crossing.
A fourth block can only take freedom away
A rod on a double slider can carry a third block on a third slide through the crossing and still turn, and a slide turned slightly wrong leaves it free only in two windows a turn. The hope was that a fourth block, at the point of the rolling circle opposite the third, would travel in opposition and cancel the error. It cannot, for two reasons. The opposite pin is the other end of a diameter: it runs on the perpendicular slide, a quarter of a turn out of phase, so its windows fall exactly between the third's. And whatever its place, a block only adds a condition the rod must also meet, so the rod's freedom is an intersection and can only shrink. With the double slider's own blocks exact, the third and the opposite fourth turned 0.0066 rad each lock the rod, where the third alone still moves at any error.
Fit every extra slide at one pose
A double slider's extra slide, turned slightly wrong, leaves its block free in a band of the slide centred on the point it was turned about, and the rod free in windows where the pin passes through that band. Turned about the crossing, a third and an opposite fourth block have their windows a quarter of a turn apart and lock the rod from 0.0066 rad. Set each slide through its own pin at one common pose of the rod and every error becomes a turn about the place its pin has then, so every window contains that pose and the rod can never lock, with two blocks or six. Each slide pivoted where it alone is freest does better while the errors are small and locks from between 0.02 and 0.05 rad, where the common pose 45° from either crossing comes within a tenth of the best pivots there are.
Teeth
A gear tooth is not a shape somebody liked. It is the curve that keeps the velocity ratio constant while the contact point slides, and there is essentially only one answer.
Why a tooth is an involute
A gear tooth is not a shape anybody chose for its looks. It is what one requirement forces — that the ratio of the two shafts' speeds stays exactly constant while the contact point slides along the flank. Impose that and the curve is essentially determined.
What happens in a mesh
Contact between two gear teeth happens only along one straight line, and only over part of it. How much of that line lies between the two tip circles, divided by the base pitch, is the contact ratio — and if it drops below one the drive periodically stops being driven.
Undercutting, and the seventeen-tooth rule
Below a certain tooth count a standard cutter eats into the flank it is supposed to be forming. The count is quoted as seventeen. The formula gives 17.097, which means seventeen undercuts slightly and eighteen is the smallest that does not — the rounding went the convenient way.
Epicyclic ratios, two ways
An epicyclic train has three shafts and one equation relating them, so fixing any one gives a different ratio from the same gears. The sign errors are notorious, so every ratio here is computed by Willis's equation and by the tabular method, and the two are required to agree.
Moving the cutter out
A gear with too few teeth is undercut by the tool that generates it, and the fix is to hold the tool further out. What that does to the tooth is easy to say. What it does to the pair is not what most readers expect — the two gears no longer mesh at the centre distance the sum of their radii would give, and the pressure angle they run at is no longer the one they were cut with.
Backlash is an allowance
A gear pair with no backlash cannot be run, and a pair with the wrong amount cannot be assembled. It is bought with a centre distance — 0.03 too far apart on a 30 mm centre buys 0.022 of it, which is a quarter of a degree at the pinion — and the textbook formula that says so is right about the slope and drifts 4.5% at a centre-distance error nobody would accept anyway.
A clock is a factorisation
A going train's job is a ratio and its parts are tooth counts, so whether a clock can be built is whether a number factorises inside the counts a wheel-cutting engine will cut. Sixty has 404 answers in two pairs and none at all in one. The ratio between a sidereal day and a mean one has none in any number of pairs, and the best two-pair train is out by a sixth of a second a day.
Which tooth meets which
A tooth on a pinion does not meet every tooth on its wheel. It meets z₂ divided by the greatest common divisor of the two counts, and it meets the same ones for the whole life of the drive. On the default planetary used throughout — sun 24, planets 24 — a planet tooth touches exactly one sun tooth and never touches another, and nothing in the drawing says so.
The module is a size, the ratio is a shape
A spur pair has exactly one length in it. Every quantity it produces is either proportional to that length or completely independent of it — centre distance, base pitch and contact length scale exactly; ratio and contact ratio do not move at all, not even by a part in ten to the fifteen.
Two flanks, one law
At a gear mesh the two tooth flanks are conjugate profiles, the pitch point is the pole of their relative motion, and their radii of curvature are tied to each other rather than free. Each flank's radius varies fourfold across the mesh and the sum of the two does not vary at all.
One rack and every wheel
Two gears mesh if they were cut by the same tool. That is not a manufacturing convenience laid on top of the geometry — it is the geometry, and it is why a gear standard is written as a description of a cutter rather than as a family of tooth curves.
A tooth flank is an unwound strand
Unwind a taut string from a circle and its free end traces the involute, to 1.5·10⁻¹⁴ mm of the curve the gears field draws. That is not a curiosity: the line of action of an involute pair is a crossed strand on the two base circles, so the property the involute is chosen for — a ratio that does not care where the shafts are — is a belt's property rather than a curve's.
The angle the standard left free
Involute geometry fixes the tooth curve and leaves one number open. Raising it buys smaller pinions and spends contact ratio, monotonically and in opposite directions, so there is no angle that is best at both — and the familiar twenty degrees is a choice with a date on it rather than an optimum.
A tooth that lives on a sphere
Every tooth in this field so far has been a curve in a plane, forced by the law of gearing and exact. A bevel tooth's profile lies on a sphere, no piece of a sphere flattens without stretching, and so the shape a bevel gear is actually cut to is an approximation — the only one in the field.
The mesh with one curvature reversed
Turn an annulus's teeth inward and the same involute law produces a different machine: a centre distance that is a difference, two base tangencies on one side of the line of action, more contact than an external pair carries, and three separate floors on the tooth counts, all of them the same statement about where an involute stops existing.
Contact that runs along the tooth
A straight tooth engages along its whole face at once, so the amount of contact at a mesh is a square wave. Slant the tooth and a second contact ratio appears that has no tooth count in it and no pressure angle — bought with face width and helix angle alone, and able to reach a value at which the total length of contact stops varying at all.
The shift that evens the wear
A pinion's flank slides hardest at its root, and on a small pinion much harder than anything on its wheel: a 17-tooth pinion on a 51-tooth wheel reads 11.1 at its root against the wheel's 1.38. Moving the pinion's cutter out and the wheel's in by the same amount leaves the centre distance alone and slides the contact path along the line of action, and at a shift of 0.357 the two roots slide equally, at 1.96. That shift is not the undercut cure — the same pinion needs 0.006 for that — and it grows with the ratio, so a pinion that needs no shift at all to be cut cleanly still needs a third of a module to wear evenly with its wheel.
Four knobs for the root
A shift of the cutter can balance the sliding at a gear pair's two roots, but on a 17-tooth pinion driving a 51-tooth wheel both then slide at 1.96. Free both shifts and both addenda, let the centre distance go where zero backlash puts it, and keep a contact ratio of 1.2, tips a quarter of a module thick and a tenth of a module of clearance: the worse root comes down to 0.83 with shifts no larger than a module, and 0.75 without that limit — two and a half times less. Every optimum has its two roots exactly equal, runs at an operating pressure angle of 24° to 29°, and sits on its contact-ratio floor, which is what the gain is paid with.
A helix pays for the root with face width
A spur pair's least worst-root sliding, 0.834 on a 17-tooth pinion driving a 51-tooth wheel, was bought by spending contact ratio down to the floor of 1.2 that continuity needs. A helical pair gets its continuity from the overlap along the face, so its transverse contact ratio can fall below that floor, and the root's sliding falls with it, nearly in proportion: 0.209 at a third of the contact and 0.102 at a sixth, on the pitch point's law π εα (1/z₁ + 1/z₂) / tan αw. The helix angle itself buys under a tenth. What pays is the face: the mean length of line in contact is the transverse ratio times the face, so carrying the spur pair's load on as much line at a third of the contact takes a face 2.65 times as wide.
The face carries the least line
A helical pair that gives up transverse contact to cut its root's sliding was priced by the face it needs to carry the spur optimum's mean length of contact line: 2.65 times as wide, at a third of the contact on a 30° helix. But the length in contact swings through every base pitch and the peak load stands on its least value. That least has a closed form in the fractional parts of the two contact ratios, exact against the swept field to 10⁻¹³. The spur optimum's own least is its face, a sixth below its mean, so least against least the factor is 2.21 when the overlap ratio is a whole number. Real faces land between whole numbers and need 2.21 to 2.87 times the spur's, 2.39 on a spur face of ten modules, and above 2.65 only on faces under ten. And a steeper helix is not always more line: at a fixed face, raising the overlap ratio from 1 to 1.5 takes a third of the least line away.
The least line carries the root
A helical pair's length of contact line swings through every base pitch, and the peak load stands on its least value. The roots, where the sliding is worst, sit at the two ends of the path of contact. The share of the load carried there swings too, and in the limit of a thin root band it reaches its peak while the line in contact is least: half as much again as an even share at overlap ratios just above one, a quarter above two, a sixth above three. A whole-number overlap ratio removes both swings at once. So the overlap that makes the least line largest is also the one that spreads the root's load evenly, and a face sized on the least line alone understates the root's worst moment by the root's own swing.
Prescribed motion
A linkage gives the motion its geometry allows. A cam gives the motion you asked for — and the cost is paid in accelerations you did not ask for.
Prescribing motion
A linkage gives the motion its geometry allows. A cam gives the motion it was asked for, which sounds like an improvement and is a trade — the displacement becomes free and the derivatives stop being.
The law that costs least is not the smoothest
Constant acceleration gives the lowest peak acceleration of any motion law and an impulsive jerk. Cycloidal motion has finite jerk everywhere and a peak acceleration 57% higher. The trade is real, it is measurable, and the displacement curves that everyone plots give no hint of it.
Stopping thirty times a second
A Geneva wheel turns continuous rotation into steps, and its one design requirement is that the pin enters the slot along the slot so the driven wheel starts and stops from rest. That fixes every dimension from the slot count. What it does not fix is the acceleration, which is why film sprocket holes tear.
The cam that cannot be cut
A cam has to be big enough for two reasons, and they are not the same reason. One is that the follower will jam in its guide if the pressure angle is steep. The other is that the roller will gouge the profile if the curvature is tight — and there is a combination where the pressure angle is comfortable and the cam still cannot be manufactured at all.
A profile is an envelope
A cam's surface is not a curve somebody drew. It is the envelope of the roller as the roller runs round the pitch curve, and its curvature is the pitch curve's less the roller radius — a law that the cam calculations here were breaking for six months, in the one curve that gets manufactured and the only one no check looked at.
A cam is a conjugate pair
A cam is built by offsetting the path of the follower's centre inward by the roller radius. It can also be built by asking what shape stays in contact with a circle that slides in a stated way — the same computation that cuts a gear tooth — and the two surfaces agree to sixteen millionths of a millimetre.
A follower needs a face
A flat-faced follower does not touch the cam on its own axis. The contact wanders sideways as the cam turns, by exactly ds/dθ, and a face cut to the lift or to the base circle or to whatever looked right is a face the cam runs off.
A lift is a size and a law is a shape
A cam's motion law is a dimensionless function of a dimensionless argument, so everything the cam field says about laws — which has the least acceleration, which has an impulsive jerk, which one costs least — transfers between cams of any size. What does not transfer is anything with the lift or the base circle in it.
An offset trades the rise for the return
Moving a roller follower's line of travel off the cam's centre lowers the pressure angle on one stroke by raising it on the other. When the rise and the return are mirror images the best offset is zero. When the cam rises in 90° and returns over 170°, an offset of 5.88 takes the worse stroke from 27.37° to 21.81° by handing the return 6.48° it did not need.
A flat face asks for a convex cam
A flat-faced follower is pushed at right angles to its face, so its pressure angle is zero at every angle and the constraint that sizes a roller's cam disappears. What replaces it is convexity: the profile's radius at the contact is R₀ + s + s″, and a base circle below 5.332 leaves a cycloidal cam with a hollow the face cannot reach — held 0.446 high on a base of 2.
A cam that holds its follower both ways
Two parallel faces joined into a yoke can drive a follower out and back with no spring, but only if the cam's breadth along the follower's line is the same at every angle — s(θ) + s(θ + 180°) constant. That makes the second half-turn the first one reflected. The cam field's standing programme misses by 9.502, and a second disc that frees the programme needs a yoke at least 60.66 wide.
An arm is an offset that grows with the lift
Carry a cam's roller on a swinging arm instead of a slide and its pressure angle obeys the offset follower's formula exactly, with the offset replaced by the distance of the roller's line of motion from the cam's centre. That distance turns with the arm. On a cam whose strokes are mirror images the best arm is worse than a centred slide by about 5,000/L² degrees and leans the follower by its own tilt on the dwells. On a quick-rise cam the pivot's angle rebalances either side of the cam for under a degree, where a sliding offset moved to the wrong side costs eleven.
A roller in a groove changes walls with the speed
A groove holds its roller between two walls and chooses between them by the sign of one force, the follower's mass times its acceleration plus the load pressing it in. Below a threshold speed that force never changes sign and the roller stays on the inner wall; each stretch of deceleration adds two crossovers above its own threshold. On a quick-rise cam at 20 N and half a kilogram the count goes from none to two at 268 rpm and to four at 506. A cycloidal law brings the roller across smoothly; a constant-acceleration law throws it across by a step of 3,200 N at 3,000 rpm.
A flat face on an arm is worse
Carry a cam's roller on a swinging arm instead of a slide and the pressure angle improves — a long arm beats an offset. Carry a flat face on one and the opposite happens. The cam sees the sine of the follower's rotation rather than the rotation, and the distortion costs convexity: the smallest workable base circle rises from 10.66 on a slide to 28.68 at a pivot three base circles out, and below a certain arm length no base circle works at all.
The time a crossover takes
A roller in a groove changes walls where the groove's force changes sign, and it gets there by flying across the clearance. How hard it lands depends on the clearance through an exponent the motion law decides — two thirds where the force passes through nought, one half where it steps — a flight that ends in a dwell lands at √(2cF/m) whatever the speed, and just above each threshold the force changes sign and the roller never arrives at all.
Where on the sine the arm swings
A flat face on a swinging arm delivers its programme through d sin ψ. Differentiating that twice brings in a square of the follower's speed, multiplied by sin ψ, and sin ψ has a sign. Standing the face off its pivot moves the swing down the sine until it straddles ψ = 0, where the square helps on one half of the stroke. On an arm ninety out that lowers the smallest convex base from 28.68 to 26.14, and it lets an arm of 61 have a cam where the face through the pivot needs 80. It never reaches the sliding face's 10.66.
A cam behind a bent joint
A cam's motion law is written in cam angle and delivered in time, and the two are the same only on a shaft that turns steadily. Drive the camshaft through a Hooke's joint and the follower's acceleration gains a second term — the law's velocity times the shaft's angular acceleration — that a steady shaft does not have. Where the rise falls in the joint's half-turn then decides everything: at a 30° bend the same cycloidal rise peaks 35% above its steady value or 17% below it. And past a bend of 34.5° a well-clocked cycloidal cam gives its follower less peak acceleration than the constant-acceleration law, the one designed to minimise it.
A cam on a chain steps at every tooth
A roller chain leaves its sprocket at a radius that shrinks and recovers as each pin seats, so a chain-driven camshaft speeds up through every tooth and its angular acceleration jumps back at every seating. That puts a step into the follower's acceleration once per tooth, whatever the law — a cycloidal rise, chosen for its continuous acceleration, gets four steps inside a 120° rise on an eleven-tooth sprocket, each 0.41 of its peak. The harm falls with the tooth count, as one over it, not up as its square, and the simple harmonic law, whose peak sits where its velocity is nought, is hurt least of all.
An arm that lifts against the cam can beat a slide
A flat face on a swinging arm seemed to offer three choices: how far its pivot is from the cam, how far the face stands off the pivot, and the angle the face makes with its bracket. The angle is not a choice. A line fixed to a body turning about a pivot is fixed, up to the turning, by its distance from the pivot, so a face tilted β on a bracket c long is the face on a parallel bracket c cos β long, the same line at every instant. The real third choice is which way the arm lifts. On a symmetric programme the two ways are mirror images. On a quick rise, an arm lifting against the cam's turning needs a base circle of 6.6 to 13 where a sliding face needs 32.85, and one lifting with it needs 41 to 59, or has no convex cam at all.
A steep rise spends the arm's room
A flat face on an arm that lifts against the cam's turning needed a fifth of a sliding face's base on a 90° rise. Steepen the rise to 45° and the slide's base grows 5.6 times, to 185.6, while the arm's least base grows only 3.2 times, to 23.3. But the arm's least base sits where its cam is about to go concave on the return as well as the rise: the base can be raised by only a hundredth before the return fails. Ask for a tenth's room above the base and the arm's base grows 5.1 times, nearly the slide's rate, and at 35° no stand-off gives that room at all.
A gentle return cannot widen a steep rise
An arm lifting against a cam on a 45° rise had its least base pinned by the return: raise it a hundredth and the return went concave. Give it a return chosen for the arm, slower or gentler, and the least base falls from 23.3 to 18.1, a fifth less. But the window above it only grows from 1% to 4.9%, because once the return is out of the way the rise closes the window itself, and the arm's own reach closes the next one. With a tenth's room the best return saves 8% of base, and no return of nine gives a quarter's room. Constant acceleration, the law with the lowest peak deceleration, is the worst return of the three.
Drawn wrongly
The mechanisms that are illustrated confidently and incorrectly, the ratios quoted from the wrong formula, and the pictures that would not move if they were built.
The ratio that is not a number
A four-bar's output-to-input speed ratio runs from −0.29 to 0.51 through one turn and changes sign on the way. Quoting a single figure for it quotes the average of that curve, which the mechanism never exhibits. A gear pair is the case where the same phrase is honest, and it is honest by construction.
Two things called jamming
A four-bar's mechanical advantage peaks at 1,673 in one configuration and its transmission angle collapses in another, 222° away. Both get described as the mechanism jamming. One is enormous force output and the other is force disappearing into the bearings, and they are opposite situations.
Exactly right, and unbuildable
A linkage synthesised through three prescribed positions reaches all three. That is a theorem and it holds exactly. Whether it reaches them in one piece, without being taken apart, and in the order asked for, are separate questions the construction says nothing about — and of 1,176 exactly correct solutions, 176 could be built.
The solver was refusing a quarter of the sweep
Four numbers in the Jacobian these essays used had the wrong sign, from the first spatial essays until now. Every picture it ever drew was correct, because a wrong derivative does not move a converged answer — it just makes Newton crawl, until the stall rule declares the position unreachable. The symptom was a sweep quietly returning fewer frames than it asked for, and no gate in the fleet has a rule against that.
The escapement that could not alternate
Four drawings of intermittent mechanisms that appear everywhere and would not work: pallets spanning a whole number of teeth, a pawl whose pivot is on the wrong side of the tooth face, a Geneva at the wrong centre distance, and an intermittent gear with no locking arc. Each one is put through the library that draws the working version, and each returns a number.
The gearset that could not be assembled
A planetary drawing shows a sun, a ring and three or four planets between them, and if the circles are the right sizes at the right stations it looks right. The condition that decides whether the second planet can actually be dropped in is arithmetic — the sun and ring teeth must add to a multiple of the planet count — and it appears in no drawing, at no scale, in any style.
Not unreachable, only expensive
The sentence is false of every wheeled mechanism in this field and true of exactly one — the trolley bolted to a rail. A rolling constraint forbids a direction and reaches everywhere; the mistake is reading a statement about instants as a statement about intervals, and it is made in both directions.
Six things a centre is not
The instantaneous centre is the most over-read object in this subject. Six claims about it are in circulation, three are false, two are true of something else, and one is nearly right — and each of them comes with a measurement of how badly it goes wrong.
Six things a shape is not
A rotor that is not a Reuleaux triangle, teeth that are not the same shape as each other, a fillet that is not an arc, a mesh that does not roll, a conjugate pair that cannot be built and a tooth form that was not deduced. Six claims in circulation, each with the number that kills it.
Six things a strand is not
A tackle that is not four to one, a wrap that is not a half turn, a tensioner whose travel takes up nothing, a winch speed that is not a property of the winch, a cable rig that holds nothing still, and a shaped pulley that cannot be asked for what it is usually asked for. Six claims, each with the number that kills it.
Six things a network is not
A count that is right about a difference and read as an answer, a nullity taken for a mobility, a flat state that cannot tell a mechanism from a structure, a scissor ring that closes nowhere, a vertex that folds while its sheet does not, and a null space computed with an instrument whose floor is above the answer. Six claims, each with the number that kills it.
Six things a hold is not
A rank read as a restraint, a count read as an answer, four contacts placed the wrong way round, a nullity taken for a spin, a part free in every direction and unable to leave, and a tolerance offered as a cure for an arrangement that was never a hold. Six claims, each with the number that kills it.
Six things a chain is not
A count read as a verdict, a rank trusted where it is blind, a fingerprint used as a proof, a list of five taken for a complete one, a solver treated as a convenience, and a census read as a catalogue of machines. Six claims, each with the number that kills it.
Six things a joint is not
A freedom count read as a description, a screw system read as a group, a pair list read as a convention, a trajectory read as a determination, a nominal alignment read as a delivered one, and a higher pair read as a larger joint. Six claims, each of them what a careful person would say, each answered with a number.
Six things a compiled linkage is not
A closure residual read as a verdict, a theorem read as a design, an exact answer read as an accurate one, a degree read as a cost, a construction read as a search, and a neighbourhood read as a turn. Six claims, each of them what a careful person would say, each answered with a number.
Six things a body is not
A verdict read as a measurement, a hull read as a part, a sweep read as a proof, a drawing read as a configuration, a geometry read as a force, and a plane read as a place. Six claims, each of them what a careful person would say, each answered with a number.
Six things a measurement cannot tell you
A calibration with a perfect residual whose fourth number is a starting guess, a rank that says nothing about a second answer, an improvement that proves nothing about a parameter, a class with no margin, a plan scored on poses that were refused, and a model missing something no data can find. Six claims, each with the number that kills it.
Six things a power split is not
A multiplier of its variator's span, a machine whose neutral the belt decides, a source of any ratio near that neutral, a travel that slides towards its pole for free, a design whose attachment is the choice that matters, and a machine a second planetary always improves. Six claims about continuously variable transmissions built as power splits, each of them what a careful person would say, each answered with a measured number.
Six things a cam drive is not
A follower that is better off offset, a flat face with no size limit, a swinging arm that improves any follower, a groove that keeps its roller on one wall, a smooth law that is gentle everywhere, and a camshaft that delivers the law it was cut to. Six things people say about cam drives, each the natural reading of something true, each answered by a number the cam essays measured. They fail the same way: each reads the cam at first order, in positions and slopes, and the drive is decided at second order, in accelerations and curvatures.
Six things an indexer is not
More pins make a Geneva smoother, a finer ratchet is a more precise one, a locked wheel is a still wheel, a curved slot gives a smooth start, a cam indexer is always gentler than a Geneva, and a mechanism whose states are all in one piece can go from any of them to any other. Six things said about indexing mechanisms, each the natural reading of something true, each answered by a number the timing essays measured. They fail the same way: each judges the mechanism by its motion, and an indexer is decided at its switches — the instants a pin enters, a pawl drops or a lock takes over.
Six things a balancing spring is not
More springs share the load, a spring that must push wants a compression spring, a stiffer spring balances better, a refitted stiffness repairs a free length, least squares finds the balance, and the best spring is one of no free length. Six things said about springs that balance a mechanism, each the natural reading of something true, each answered by a number the essays on the desk lamp and the rocking slide measured. They fail the same way. Each treats the spring as the thing to be sized, and a balance is decided first by where the spring is hung and how its error is judged.
Out of the plane
In space a body has six freedoms and a pin takes five away, so a closed loop needs seven joints before it moves at all. The mechanisms that move with four are not curiosities — one of them is in every car ever built.
Six freedoms, not three
Every mechanism in these essays so far has been flat, and flatness is not a simplification made for teaching — it is a special case that hides the most interesting thing constraint counting does, which is get the answer wrong about mechanisms that are in daily use.
The joint that is not constant velocity
A universal joint is the spatial mechanism everybody has met and almost nobody has been told the truth about. Its output shaft runs fast, then slow, twice per revolution, and the amount depends only on the angle between the shafts — which is why cars have two of them and why the second one has to be fitted the right way round.
Sarrus, and the straight line that is exact
The planar answer to the straight-line problem took two hundred years and arrived as an inversion cell with eight bars. There is a six-bar answer that is also exact, that was published eleven years before Peaucellier's, and that works for a reason with nothing to do with inversion — it leaves the plane.
Bennett, and the condition that moves it
A spatial four-bar is immobile by every count there is, and generically it cannot even be assembled at more than isolated configurations. Bennett found the one relation between four lengths and two twists that makes it turn through a full revolution — and break the relation by two parts in a thousand and most of the travel is gone.
Every motion is a screw
Chasles showed that any rigid displacement whatever is a turn about some line together with a slide along that same line. Not approximately, and not usually — always, with the line and the amount of slide computable from the motion. It is the fact that makes spatial kinematics a subject rather than a pile of special cases.
What a mechanism cannot do
A mechanism's freedoms are a subspace of screw space. Everything orthogonal to that subspace under the reciprocal product is a force the mechanism carries without moving — so the constraints are not a separate thing to be worked out, they are what is left, and one matrix gives both.
Two ways to be overconstrained
A planar four-bar and Bennett's four-bar report the same redundancy, the same rank and the same wrong count. One of them is overconstrained at every set of link lengths; the other at exactly one ratio and nowhere near it. The difference is not in any of the numbers so far — but it is measurable, and the measurement is an angle.
When the link lengths are angles
Put every axis of a four-bar through one point and the mechanism lives on a sphere. Its bars become arcs, its lengths become angles, and every planar result carries over with a sine where a length used to be — including Grashof's condition, which still predicts exactly which link goes all the way round.
The smallest screw system has a shape
Add two screws together in every proportion. The results do not scatter — their axes sweep a ruled surface, with the pitch varying along it between two extremes reached at right angles to each other. It is a picture nobody would guess from the algebra, and it is the object that says what two joints between two bodies leave free.
Twelve bars and a symmetry
Six joints and twelve bars in space is Maxwell's count exactly: no mechanism, no redundancy, nothing spare. Place three pairs of the joints so that a half turn about one line exchanges them and it moves — a finite motion, walked with every bar held to five ten-thousand-billionths of its own length, on a framework the arithmetic calls a structure.
Bennett's condition is a ratio
A spatial loop's parameters are lengths and angles together, so a scaling touches only half of them. Bennett's condition — a over sine alpha equals b over sine beta — is a relation between the two halves, and what it demands of a machine is a relation between its lengths and its twists rather than a property of either.
In space there is one chain
A body in space has six freedoms and a revolute joint takes five, so a mobility of one needs (6n−7)/5 joints — an integer only when the link count leaves a remainder of two on division by five. At seven links every link is binary, the graph is a single seven-cycle, and there is exactly one spatial chain.
A name for each overconstraint
The spatial field separated subgroup overconstraint from paradoxical by measuring how far a mechanism's screw system turns: 2 × 10⁻⁶ degrees against 89. That is a verdict without a name. Closing the logarithms of the reached displacements under the bracket gives the same verdict and says which group — planar, spherical, a translation — and for Bennett's linkage it says six.
What a leg of three joints leaves free
Five essays of this field have computed the order of a screw system and drawn none of them. A three-joint leg spans a three-system; its three principal axes are mutually perpendicular and meet at a point, six numbers price every screw in the family, and the directions of the lines it contains form a cone.
The lines a leg turns about and the lines it is pushed along
A leg of three revolute joints permits a three-system of motions and resists a three-system of loads. The two share a centre and three axes and differ only in the sign of every pitch — and the revolute axes of the first and the lines of force of the second are the two rulings of one hyperboloid, every line of one meeting every line of the other.
A shared load is a shared motion
Put two legs of three revolute joints under one platform and let them share a line of force, so that both carry the same load. The platform does not merely gain a redundant load path; it gains a motion in the same act, because three constraints and three make six. The motion is a screw about a different line, tied to the shared one by h cos φ = d sin φ on two hundred constructed pairs to 4 × 10⁻¹⁵. Pushed along it, the two legs part as the square of the push: the platform shakes and does not move.
A third leg that keeps the shake
Two three-joint legs that share a line of force share one twist as well, and a third leg in general position takes it away. A third leg whose own freedom contains that twist does not, and building one is three conditions on twelve numbers: two joints anywhere and the third solved for. Such a leg must share a load with each of the first two — but not the line they share. What it shares with each is a screw with a pitch of its own, reciprocal to the twist through the sum of their two pitches, and only a third leg built through the pair's line shares the line itself. With any of them the platform still shakes rather than moves.
A shake has an order
Two legs of three joints that share a line of force let a platform shake: pushed along the one twist they share, the legs part as the square of the push. The square's coefficient is a bracket of the legs' own joints, computable without pushing at all, and it can be tuned to nought by turning one joint. The shake does not become a motion. It moves up an order, to the cube, and a second tuned joint moves it to the fourth power. Each order is one more condition, and only legs whose freedoms are groups meet them all.
A shake travels as a root of the play
Two three-revolute legs that share a twist let their platform shake along it: the gap between their nearest poses grows as the square of the push, or the cube once one joint is tuned, or the fourth power once two are. With play in the joints, the pair moves until that gap uses up what the play can take, and that is a root: the square, cube or fourth root of the play. At a play of a millionth the three pairs travel 0.013, 0.066 and 0.139 along their twist. A tuned joint is worth more the tighter the joints are, five times the travel at a millionth and eleven at a hundred-millionth.
Three tuned joints stop at the fifth order
Two legs of three revolute joints that share a line of force let their platform shake, and every joint of the second leg turned about that line clears one more order: the square of the push, then the cube, then the fourth power. A third tuned joint was expected to buy the fifth. Read off the two legs' own Taylor series instead of a push test, the set where the second- and third-order coefficients vanish is a curve in the three tuning angles, and the fourth-order coefficient crosses nought on it twice. At both points the gap falls as the fifth power of the push, confirmed independently with slopes of 4.97 and 4.89. The fifth-order coefficient is −0.0042 at one and 0.109 at the other, and with three angles spent there is nothing left to clear it: three tuned joints stop at the fifth order. And a higher order is not always more travel: in a millionth of play the second of them moves less than the fourth-order pair.
Several legs, one platform
Everything else here has one path from the ground to the moving part. Give it three, and the easy problem and the hard one change places — and a new kind of singularity appears in the middle of the workspace, where the machine can move with every motor locked.
The easy problem and the hard one change places
For a robot arm, working out where the hand is takes a walk down a chain and working out what the joints must be to put it somewhere is the hard part. Connect the platform by three legs instead of one and both statements reverse — and the reversal is not a matter of degree.
Locked, and still moving
A serial arm goes singular at the edge of its reach, where it loses a freedom, and the failure is visible as an arm gone straight. A parallel mechanism has a second kind with no serial counterpart — it gains a freedom, in the middle of the workspace, at poses nothing about the legs' reach marks out.
One command, six answers
Lock the three motors of a planar platform and the platform can be in as many as six different poses, every one of them satisfying every leg exactly. Which one it is in was decided by how it was assembled and where it has been since — and the number of answers is not a property of the mechanism but of where the motors happen to be.
Six legs and a square root
A Gough–Stewart platform's inverse problem is one subtraction and one square root per leg, computed six times without any leg consulting another. Its forward problem has forty solutions. That gap is the whole design, and it is why flight simulators are built this way and robot arms are not.
Why the platform stays flat
A delta robot has three legs and three freedoms, and there is no obvious reason those freedoms should be the three translations rather than some mixture. The reason is a parallelogram in each leg, and the argument from there to "the platform cannot turn at all" is a constraint computation that takes six wrenches and a rank.
The workspace is not a shape you choose
A serial arm's reach is roughly a sphere and can be quoted as a number. A parallel mechanism's is the intersection of three reachability conditions, changes with every degree of orientation, and has a surface of uncontrollable poses cutting through the middle of it. There is no formula. There is a map, and it has to be computed.
Three legs and one plane
The parallel field argued that a platform stays flat from the structure of its legs, one leg at a time. The same fact is one line of linear algebra: each leg confines the platform to a group, the platform gets the intersection, and an intersection of groups is a group whatever the leg lengths are. The motion type is decided before a single dimension is chosen.
A platform that measures itself
A three-legged platform with every joint read can be calibrated from its own sensors with no instrument in the room. Left to itself it shrinks the machine to a fiftieth of a per cent of its size — and reports a residual five orders smaller than the right answer's for doing it.
Round a cusp into another assembly
A parallel platform's assembly mode was supposed to change only through a direct singularity. Driven round a small loop of motor angles that encloses a cusp of the singular curve, the standard three-legged platform leaves one assembly and arrives in another, turned 52° from where it started, and at no point on the way is it nearer than 0.0716 to singular.
The smallest parallel robot
Two motors, two arms, and two links meeting at a hand: a planar five-bar is the smallest parallel robot there is. Its forward problem is two circles, so the hand has two places to be, and each is named by the sign of one determinant. That is the whole reason it cannot do what the three-legged platform does, and cannot change assembly without passing through the one configuration where the two meet.
What a fourth leg buys
Three leg lines fail to hold a platform when they meet at a point, which is one condition, so in every slice of the workspace the failures form a curve. Four lines fail only when all four meet at a point, which is two conditions, so the curve becomes isolated points. The fourth leg buys that and more, and it costs a machine that can no longer be assembled from any four motor angles.
A yaw that is singular everywhere
Turn the standard hexapod platform 30° about the vertical, hold it level, and it is singular: not at one pose, but at every position it can be put in, with one screw motion that none of its six locked legs can resist. The angle is not a property of the dimensions. It comes out of one line of trigonometry that no spread of the anchor points can change.
Two orientations no position can rescue
A planar platform on three extending legs whose platform triangle is a scaled copy of its base is singular at every position it can be put in, at exactly two orientations: its three leg lines meet at one point wherever it is. The two orientations are read off the attachment points, a platform a few per cent from similar is held at its worst orientation only in proportion to how far from similar it is, and the revolute-legged 3-RRR, built on the same similar triangles, does not inherit any of it.
One placement of every placement
A planar platform whose two triangles are similar is singular at every position it can be put in, at two orientations. A fourth leg removes both — unless its own pair of attachment points is related by the same similarity, and then it changes nothing at all. Swept right round the platform, exactly one placement of the fourth attachment fails, the similarity names it in advance, and the holding a placement buys is proportional to how far it sits from that one point.
The dead yaw is a design choice
A paired hexapod held level is singular everywhere at a yaw of 30° because its platform pairs sit 60° round from its base pairs. Rotate them by ρ instead and the dead yaws move to 90° − ρ and 180° from it, exactly, at every rotation from 0° to 120°. The furthest they can be from home is a quarter-turn each way, at ρ = 0 — where the platform is also best held at home and can turn ±75.5° before its holding halves, against ±17° for the usual 60°.
Tilted, near the dead yaw
A paired Gough platform held level is singular at one yaw wherever it stands. Tilt it and that stops being true — the six moments are no longer equal and the home position is held. It is a poor rescue: the rise is quadratic in the tilt, so a degree buys a sixty-fourth of what eight degrees buys, and what the tilt actually does is not remove the singularity but turn it into a surface through the workspace.
The dead yaw opens into a wedge
A paired Gough platform held level is singular everywhere at one yaw and nowhere a quarter of a degree either side of it. Tilt it and each position of the workspace becomes singular at its own yaw, displaced from the dead one in proportion to the tilt and to the position's horizontal offset over its height. So the orientations that put a singularity somewhere in the workspace are not a surface through the dead yaw but a wedge standing on it, about a quarter of a degree of yaw wide on each side for every degree of tilt — and the rotation between the pairs, which chooses where the dead yaw is, barely changes how much room it needs.
No path across the dead yaw misses it
A paired Gough platform that has to turn from one side of its dead yaw to the other cannot be steered round the singularity by tilting, by moving, or by turning the long way. The six legs' determinant has one sign everywhere on one side and the other everywhere on the other, so every path passes a singular pose — two hundred random ones do, an odd number of times each. Tilt chooses only where the crossing happens and how briskly: a quarter quicker at best, and threefold slower if it is done by moving instead of turning.
A seventh leg opens the dead yaw
Six legs can never carry a paired Gough platform across its dead yaw without passing a singular pose, because their determinant changes sign there and every path crosses nought. Seven legs make no square matrix and have no determinant to change sign. They fail only where the twist the six leave free also does no work on the seventh, a second condition that paths miss. Placed well, a seventh leg holds the platform's smallest singular value near 0.029 through the dead yaw, above the six legs' own value ten degrees off it, on every one of a hundred random paths across. What it cannot do is lift the platform further than the six's next-weakest direction allows.
A seventh leg must lean its own way
Seven legs on a Gough platform fail only where the twist the six leave free does no work on the seventh. Near the paired platform's dead yaw that set is decided by one angle: how far round the axis a leg's platform attachment sits from its base attachment. The six legs all lean 82.5° or 97.5°. A seventh at either lean joins them and is singular at every level position; a seventh at any other lean is singular at none. Tilted, the failing leans spread out from the six's, to 74.5°–107° at four degrees of tilt and 44.5°–137.5° at eight, and never reach the far side of the axis, where the best leg leans.
A seventh leg has no dead yaw of its own
Seven legs can only be singular where six already are, since a 7 × 6 matrix that loses rank takes every 6 × 6 minor with it. Level, over a whole turn of yaw and the whole workspace, the paired Gough platform's six legs are singular at two yaws only, the dead yaw and the yaw half a turn on, and at both a seventh leg fails exactly when it leans as the six do. The best seventh leg leans the other way at both, so it has no dead yaw of its own. Its singular set exists, and it hugs the six legs' dead yaws, but it comes in only at tilts of 15° and more.
How many answers
Newton finds a configuration. It cannot tell you how many there are, and a search that has stopped finding new ones is not a proof that there are no more. Written as polynomials, a mechanism's closure conditions have a number of solutions that is a property of its shape — and this is where the site stops counting by looking.
Two circles, four answers
A four-bar with its crank held still is two circles, and two circles meet twice. Bézout's theorem says four. The two missing answers are not a rounding error and are not special to these link lengths — they are the same two points for every pair of circles ever drawn, and they are the beginning of a way of counting that has so far been done by hand.
Following a root from a problem already solved
Homotopy continuation solves a system nobody can solve by deforming one that anybody can, and following every root as it moves. The whole method rests on the deformation being generic, and the folklore says that is what the γ-trick is for. Running all four combinations says the folklore names one of two places the randomness can live, and either will do.
The paths that leave
Bézout's number over-counts, and the over-count is enormous — 1,458 tracked paths for 80 solutions. The obvious response is to find a method that tracks only the paths that arrive. That method exists, it was built, and it is four times slower, because the surplus paths are not merely surplus. They are cheap.
The count that does not move
A mechanism does not have a number of assembly modes. Its family has a complex solution count that never changes, and each member has a real count that does — 676 sets of leg lengths for one platform, all with six complex solutions, and nought, two or four of them real. The number a machine shop cares about is the one that is not a property of the machine.
Twenty-eight, not forty
The general six-legged platform has forty poses for a given set of leg lengths, and that number has been quoted beside a picture of a platform that has twenty-eight. Its anchors are arranged symmetrically, which makes it a special architecture, and the missing twelve poses are not missing. They are at infinity, and perturbing the anchors brings them back.
The search that was right
This site has reported sixteen assemblies for its Gough platform and labelled the number a lower bound found by search, everywhere it appears. Tracking every path says there are twenty-eight poses and sixteen of them are real. The lower bound was tight. Nothing available to the search could have said so, and a second method that was supposed to settle it turns out to have the same defect one level up.
Two to the power of the dyads
How many ways a mechanism can be assembled at a given input angle is a count of configurations, and it is predicted here by a graph: two circles per pair of links, so two to the power of the number of pairs. The prediction came back four for Watt's chain and the count came back eight, and the four extra had a link turned inside out.
The equation a four-bar satisfies
Every textbook says a coupler curve is a sextic. Traced at six hundred solved positions and fitted at degrees four through seven, the answer comes back as a measurement: nothing vanishes below six, degree six drops by ten decades, and degree seven buys nothing — with the decision made by a gap of 2.8 × 10³ rather than by a residual.
Never three circuits
A four-bar has one circuit or two, and twenty thousand random four-bars counted exactly contain no exception. The reason is a count of four points where the two assemblies merge, which makes the configuration curve a curve of genus one, and Harnack's theorem allows a real curve of genus one two pieces and no more. A six-bar's curve has genus five or seven, and its circuits go to four and six.
A degree counted on a line
A curve of degree six meets a general line in six points, and that sentence is a way to measure the degree with no equation in it. Written as polynomials, a four-bar and a random complex line have eight paths to track; six arrive on every line tried, the other two run off towards the circular points, and a sum of the six stays straight to fifteen figures only when none is missing.
The curve nobody eliminates
A point on the arm of the site's dwell six-bar draws a curve whose equation nobody writes down and no fit can find: at degree eighteen a fit needs a hundred and ninety coefficients, and its singular values have no gap to decide by. Sliced by a random line, the same machine has thirty-two paths to track and eighteen arrive, on every line tried, all eighteen distinct and all on one curve.
Nine times through each circular point
A line through a circular point meets a curve at that point once for every time the curve passes through it, so counting its finite meetings measures how often, with no equation. The Stephenson six-bar's arm curve, degree eighteen, meets such lines in nine finite points: it passes nine times through each circular point. Every curve drawn by a body of a machine built from pins alone does this at exactly half its degree; a slide breaks it.
Every rational gear ratio has a degree
Mesh a gear on each crank of a five-bar and the pin where its couplers meet draws a closed curve at every rational ratio. Its degree is 6 at 1 to 1, 16 at 3 to 2, 68 at 13 to 8 and 178 at 34 to 21: four times the larger term of the ratio plus twice the smaller, read off the span of one polynomial in one variable. Along the golden ratio's convergents the degree grows by the golden ratio at each step, and at the golden ratio itself no fit finds any.
The mesh inside keeps the half
Mesh a geared five-bar's two gears inside each other instead of side by side and the cranks turn together rather than against each other. The curve its pin draws has exactly the same degree at every rational ratio — and it passes through each circular point half that degree, which is as often as any curve can, where the counter-rotating machine manages between a third and two fifths. The sign of the ratio is the whole difference.
Where a slide puts the rest of the degree
A slider-crank's connecting rod draws a quartic that passes once through each circular point, which leaves two of its four meetings with the line at infinity unaccounted for. They are not along the slide. A point u along the rod and v across it sends them to the complex slopes [2v ± i(1 − u² − v²)] / [(1 + u)² + v²], whatever the crank, rod or offset — confirmed by slicing and by the fitted equation — and they are real only for points exactly a rod's length from the crank pin, where they merge into one direction at half the point's angle.
The double points that leave for infinity
A slider-crank's connecting rod draws a quartic that owes two double points. Close to the crank pin they are two real crossings. Between a crank's length and a rod's length from it they are a complex pair and the curve crosses itself nowhere. At exactly one rod both run off to the same real point at infinity and meet there as a tacnode whose branches are complex. Past that they come back real. Offsetting the slide changes the two circles by (R² − a²)(R² − b²) + e²R² = 0 and sends only one of the pair to infinity.
The double point a slide cannot keep
A four-bar's coupler curve owes three finite double points and a slider-crank's owes two, so a four-bar whose rocker grows into a slide has to lose one. Parity decides which: the one that leaves is always real, and always an isolated point no configuration reaches. It runs off along d(|w|² − 1)/(w̄ − 1) times the rocker's length, a law that knows nothing of the crank, the rod, the offset or where the rocker touches the slide. On the circle one coupler length from the crank pin the law is zero, and all three double points leave together like the cube root of the rocker.
The trammel takes both double points
Grow a slider-crank's crank without bound and its crank pin's circle flattens into a second slide: the machine becomes the elliptic trammel, whose curves are conics with no double points at all. Both of the slider-crank's double points have to go, and parity cannot say how. They go together, symmetric through the crank's pivot, at ±i(|w| − e^(i·arg w))/√(1 − |w|²) crank lengths — a law that knows nothing of the rod or the slide's offset. Inside the one-rod circle they are two real isolated points, outside it a complex pair. And on the circle with the rod as its diameter, where the trammel draws straight lines, one of them does not leave at all: it settles on the crossing of the two slides.
A four-bar becomes a trammel along a family of laws
A four-bar becomes the elliptic trammel when its crank and its rocker both grow without bound, their pivots receding at right angles, and its three double points must all leave. Taken one link at a time the limit sends one point off along the four-bar's law and then the other two along the trammel's. Taken in one step it does something neither order sees: all three leave together, in proportion to the machine, at positions set by the coupler point and by λ, the rocker's length over the crank's. The two one-step orders are the family's two ends, λ → ∞ and λ → 0, and at equal rates the coupler point (0.3, 0.2) has its three double points up to 0.86 crank lengths from where either order puts them.
Three discs decide the real double points
A four-bar growing into the elliptic trammel loses three double points, and how many of them are real depends on the coupler point and on the ratio λ of rocker to crank. The boundary of the region where all three are real is one polynomial, R = λ⁴A + λ²B + C, quadratic in λ² and short when written in the squared distances from the two pins. Its sign is the count at every one of 23,100 cells tried. As a quadratic in λ² its discriminant is the circle with the rod as diameter times a figure of eight, so that circle is touched by every boundary and crossed by none, and a coupler point's whole history along the family is settled by three discs.
A growing four-bar reaches none of its departing double points
A four-bar growing into the elliptic trammel loses three double points, and wherever they are real the question is whether the machine passes through them. In the limit it passes through none: a configuration that reaches a point a crank's length away needs its rod along an isotropic direction, which no real rod has. How fast each point leaves the machine's reach is a ratio of four distances, √(1 + λ²)·|Z|·|Z − Zₛ| / (2b·|Z + 1|·|Z + iλ|), and it vanishes only at the pivot circles' second meeting. Every crossing is lost at a finite size: through a cusp, when the law predicts it, or by meeting another crossing and leaving as a complex pair, when it does not.
As built
Every length here has been a number and every pin a point. Give the lengths ranges and the pins holes and the answers change shape: a curve becomes a band, and some of the mechanisms this site admires most stop working altogether.
A length is a range
Every figure in these essays so far has been drawn from four numbers. No four numbers were ever cut. Give each of them a tolerance of ±0.01 and the rocker's output stops being an angle and becomes a band 0.73° wide at one part of the turn and 0.36° wide at another — and which of those a designer is told depends only on where somebody measured.
Two routes to a sensitivity
How far the output moves when a link length moves can be found by rebuilding the mechanism and solving it again, or by differentiating the constraint equations and solving one linear system. The two agree to two parts in a hundred million across a whole turn — and the second route is an independent test of the constraint Jacobian itself, whose coupler rows had carried wrong signs unnoticed.
The four lengths do not matter equally
Averaged over a whole turn, the coupler contributes 38% of a four-bar's output band and the rocker 11% — a factor of 3.4 between the ends of the ranking. A tolerance specified equally on all four therefore spends most of its money buying accuracy the mechanism cannot use, and the ranking that says so costs four linear solves.
Worst case and the square root
Four tolerances can be added as a straight sum or as a root-sum-square, and the second is smaller by between 1.42 and 1.96 through this linkage's cycle. The ceiling is √4 = 2 and no geometry can beat it. That factor is not found in the mechanism — it is bought entirely with an assumption of independence, and one fixture that locates two holes takes it straight back.
Where a stack-up stops working
On a crank-rocker the first-order tolerance estimate matches the measured band at every one of 180 positions, to eight parts in ten thousand. On a parallelogram it exceeds it by a factor of 475,512. Same arithmetic, same tolerance, two linkages that differ only in their proportions — and nothing in the calculation says which one it is being run on.
A clearance is a link
A pin in a hole is not a joint at a point. Its centre may sit anywhere within the difference of the two radii, so the two links it joins are connected by a body of fixed length and free direction — a binary link with a revolute at each end. That is not an analogy, and taking it literally makes a four-bar a mechanism with eight links, eight joints and five degrees of freedom.
How far the crank turns first
Reverse the input of a four-bar with a hundredth of clearance at each pin and the output does not move for about 2.8° of crank rotation. At the two positions where the rocker reverses it does not move at all, however far the crank is turned — the lost motion is unbounded there, and the peak in any plot of it is a property of the sampling rather than of the mechanism.
Fragility has a direction
Tilt one axis of a Sarrus linkage out of true by a thousandth of a radian and it stops dead. Tilt the same axis of the same mechanism by two hundred times as much, in the other direction, and it drives through a full turn with nothing measurably wrong. Three orders of magnitude between two errors of the same size — and the direction that matters is the one the reciprocal screw system names.
Why a hinge works
A door hinge with three knuckles is overconstrained — three axes imposed where one would do, and exactly parallel is a condition no bored hole has ever met. It works because the misfit is 0.507 times the error and the play in each knuckle is larger than that. The mechanisms these essays called unbuildable are built every day, and the thing that builds them is the clearance that was already there.
What is still outside
Eight essays here end by saying that clearance, backlash, friction or wear are not modelled. This field took two of those four, because a tolerance is a set of geometries and a clearance is a short link, and both are questions about where a mechanism can be. The other two are not, and this is the page that says exactly where the line falls and why it is where it is.
Seven lengths and a hundred corners
Nothing in a tolerance analysis is about four. A Watt six-bar has seven lengths, its corner enumeration is 128 mechanisms rather than 16, and the two routes still agree to a hundredth of a per cent — but the costs have separated — 142 solves against seven. At twenty parameters, which is an ordinary spatial mechanism, it is a million against twenty.
A band with a direction in it
One whole direction of a four-bar's tolerance box does nothing. A machine made a quarter of a per cent too big all over has an output error of exactly zero — and an aluminium four-bar heated by a hundred degrees has an output error of exactly zero, while one with a steel frame has 0.076°.
What a drop cannot be smaller than
Two thirds of an escape wheel's travel is drop, and drop does nothing. The obvious economy is to cut it down, and it cannot be cut down, because every dimension it is made of has a tolerance and a drop smaller than the accumulated error is a tooth that does not clear the pallet it is leaving. The stack is 0.39°, and it barely moves when the tooth count triples.
Which pin to buy
The four lengths of a four-bar contribute 38, 26, 25 and 11 per cent of its output error — a spread of 3.4. Its four pins contribute 32, 25, 22 and 21 — a spread of 1.5. Clearances are more evenly shared than length tolerances, because every pin joins two links and so appears in two of the four sensitivities, and that changes what a better bearing is worth.
Where the boundary moved again
The practice field's inventory ended by naming what the work after it should do first: take the feature positions as the variables and derive the lengths. That is done, and it turned out not to be an extension of the tolerance field but half of a different one — because where a length comes from and what a measurement determines are the same question.
Taking up the play
Preload does not make a clearance smaller. It takes away the clearance vector's direction, which is the property that made the error unrepeatable — so 2.80° of lost motion becomes a 0.343° offset, of which 0.136° is a constant that calibrates out. A factor of eight, bought with a permanent parasitic load that is not modelled here.
Tolerancing the holes
A drawing does not tolerance link lengths. It tolerances holes, and the lengths are derived from them — so what a length's tolerance really means depends on whether its two holes were bored in one setup. Located separately, a lengths-only stack-up is optimistic by 11%; bored together, it is pessimistic by 34%. Neither is a correction to apply blind.
The ratio that has a tolerance
Every dimension in this collection has been given a range at some point, and the ratios never were, because a gear ratio is a count and a count has no tolerance. A belt ratio is a quotient of two solved lengths, so every length in the mechanism is in it — and the amplification from belt length to ratio runs from 2.7 to 6.4 across the travel, which puts a whole per cent on a mechanism whose gearbox equivalent has none at all.
The error that is an integral
A tolerance on a link length moves an output by a bounded amount. A tolerance on a wheel radius moves a vehicle by an amount that grows with how far it has driven: one per cent of mismatch between two wheels bends a commanded straight line onto a 30 m radius, and a four-metre square comes back 1.60 m from where the machine thinks it is.
The error that is repeated
Thirty-two units cut on one setting of one machine are thirty-two copies of one error, not thirty-two draws from a distribution — so a tong's span is out by thirty-two times a unit's, not by the square root of thirty-two times it. The two estimates differ by a factor of 5.66, and the second one is the comforting one.
Where the shortest loops are
A tolerance stack-up goes round a loop, and every link the loop passes through is a dimension in it. Two eight-link chains with the same links, the same pins and the same number of loops can need twelve link lengths in their shortest independent set or fifteen — decided by the graph, before any dimension is chosen.
The pair a catalogue sells
A plain bearing is a cylindrical pair and a catalogue calls it a bearing. Add two thrust faces and it is a revolute pair, which is a different joint and changes every mobility count downstream. The kinematic identity of a bought part is decided by which surfaces touch, and the catalogue's word for it is not the same information.
Exactness a micron destroys
Lengthen one bar of a compiled machine by a ten-thousandth and its tracing point leaves the curve. On the smallest machine the error comes out smaller than it went in; on a fifty-bar one it comes out ninety times larger — and the bars that matter are the reflectors, which are the cheapest part of the machine.
Where the boundary moved
An earlier essay drew a line around what the essays here compute and listed one thing on the far side as a gap rather than a boundary: interference between links, which needed no new physics, only a body and a test. Here is what that turned out to cost and what it turned out to open.
A clearance inside a tolerance box
Every quantity derived from the lengths becomes an interval when the lengths become ranges. This one has a sign, and an interval that reaches zero is not a wider answer to the same question — it is a different answer, because on that side of it the parts do not go together.
A piano hinge is not forty door hinges
A three-knuckle hinge works because the misfit its bore errors create is smaller than the play already in its pins. A piano hinge has forty knuckles and thirty-nine of them are redundant, so the obvious reading is that it needs thirteen times the play. It needs two and a half times, and it can never need more than the bore tolerance itself — because a rigid leaf has one axis and a line through the middle of the errors misses every bore by at most the largest of them.
Machines you have met
A suspension, a steering linkage, a scissor lift, a cabinet door, a latch, a chain and a rocker arm. Every one of them is sold with a number, and every one of those numbers is a claim this site can check — which is how the field is built: take the mechanism people have actually met, solve it, and see what kind of number the catalogue was quoting.
The number on the box
Fourteen machines, fourteen quoted numbers, and five different things a quoted number can be. Six of the fourteen name a quantity the mechanism does not have at all; three are exactly right, and all three of them are counts. The field is built on the difference.
A roller is not a slider
A scissor lift has one degree of freedom, at every height and for any number of stages. Grübler's criterion agrees — if the rollers under it are counted as pins in slots. Count them as slider blocks, which is how every textbook draws a slider, and the same formula declares a machine holding a car in the air to be a structure with minus one.
Six points and no more
A ball resting on a surface is a joint: it takes one freedom away, and the force it can carry is a line through the ball's centre. Six of them, arranged well, take all six freedoms and leave a part with one place to be. Six arranged badly take five, and the sixth freedom is a screw with an axis that can be named.
The seventh contact
Add a contact to a part that is already exactly constrained and it adds no rank, so it constrains nothing — and it is the only contact in the set that can fail to touch. For four legs on a floor the combination that constrains nothing is the alternating sum of the four, which is why a table rocks about a diagonal and never sideways.
The wheel is the coupler
A double wishbone is a four-bar standing on end whose coupler carries a wheel, so camber is coupler rotation and scrub is a coupler point's path. Both are computable, and the second one comes out with the opposite sign from the model every suspension book uses — by more than the whole scrub.
A roll centre is not a point
The roll centre is a construction on the instantaneous centre of the wheel's upright, and every step of it is exact. What it is not is a height: over eighty millimetres of bump and droop it moves 54 mm on a wishbone and 131 mm on a strut, and on the strut it goes below the road.
The steering that is never right
For four wheels to roll without scrubbing, the two front wheels must point at different angles, and the relation between them is a cotangent condition no four-bar can satisfy. The trapezoid under every car meets it at straight ahead and, if the arm angle is chosen well, at exactly one other angle — 0.34° out at worst instead of 2.06°.
Locked on purpose
A toggle clamp, a landing-gear downlock and the catch on a folding table are all the same mechanism parked a few degrees past its dead centre, where the slider's motion is second order in the crank's. Seventy-one microns of slider travel undoes a latch set four degrees over — and setting it eight degrees over does not double that, it quadruples it.
Where a hinge pin can go
A cabinet door's front corner moves sideways as it opens at a rate equal to how far the pin sits behind it, so with the doors touching, no pin behind the door's face can open one without going through the next. Three millimetres of gap buys nine and a half millimetres of depth, and that is the whole reason a concealed hinge has four bars instead of a pin.
Holding an axle still
A Panhard rod moves the axle 3.56 mm sideways over 80 mm of travel and a Watt's linkage moves it 34 microns — a hundred times better, and by a higher power. The Panhard's error is quadratic in the travel and the Watt's is fifth order, which is a much stronger statement than "the Watt is better" because it says how the comparison changes with the suspension.
The chain is a polygon
A chain's pins sit on a polygon, so the radius that matters swings by 1 − cos(π/n) within every tooth: four per cent on an eleven-tooth sprocket. The quoted 53/11 is the mean of that, exactly — and how much of the fluctuation reaches the back wheel is decided by the fractional number of pitches in the taut strand, which nobody adjusts and which can take the drive from perfectly uniform to two per cent.
A ratio that is a count
A compound epicyclic's reduction is a ratio of two integers, because it is a ratio of tooth counts — 2176/106 for the drive here, exact at every position and on every unit ever made. It is the only quoted number in this field that survives being measured, and it is fragile in a way exactness does not protect against: one tooth on one ring moves it by forty per cent.
The cam is not the valve
A rocker arm's ratio is the ratio of two moment arms measured at one position, and the rocker swings twelve degrees while the valve opens. The instantaneous ratio runs 1.588 to 1.605, so the peak valve lift is 12.78 mm where the number on the box promises 12.84 — and the shortfall depends on how the rocker was set up, not on the cam.
Where the pad touches
A rocker's pad slides 1.35 mm across the valve tip through one cam event when the geometry is squared up with the valve shut, and 0.34 mm when it is squared up at mid-lift. The engine builder's rule about shimming a rocker stud is folklore that turns out to be geometry, and the same shim brings the peak lift back to what the ratio promised.
Which numbers survived
Fourteen machines were measured against the numbers they are sold with. Three survived exactly, and all three are counts. One is false by a stated bound, two are exact means of things that vary, three are honest values at a stated position, and six name quantities their mechanisms do not have. The pattern is not about honesty; it is about what kind of thing a number is.
The horsehead is a drum
An oil-well pumping unit is a crank-rocker, and the curved head on the end of its beam is not decoration. It is a drum: the cable leaves it where its face is vertical, which is the same place at every angle, so the rod moves on an exact straight line and rises by exactly the head's radius times the beam's turn. That makes the unit's torque factor the four-bar's velocity ratio, to 3 × 10⁻⁹, and leaves the crank's position as the only thing that shapes the stroke — and the direction the crank turns as the thing that decides which stroke gets the benefit.
The lamp that stays where it is put
A desk lamp that stays wherever it is left is balanced, not held: its springs store exactly the energy its weight gives up, at every pose. For one arm that takes a spring with no free length, fixed above the pivot, whose energy has a cos θ term that cancels the weight's to 10⁻¹⁵. A spring with a free length L₀ leaves an unbalance of exactly L₀/s of the weight's torque, and a stiffness a fraction ε off leaves 2ε of its moment. For two arms it takes a parallelogram: without one, the upper spring sees the angle between the arms and the lamp's energy varies by twice the head's moment; with one, both springs sit at the base and the lamp is flat everywhere.
Two springs stand in for a parallelogram
A lamp's head tilts on the end of its arm, and its weight's moment is a cosine of the head's angle from the vertical, so a spring can balance it only if the spring sees that angle. A second parallelogram carries the vertical out to the wrist and one spring does it. So do two springs with one parallelogram, or six with none: each spring that reaches across a joint brings a term in the joint's angle, and another spring across that joint takes it away. A spring across the wrist alone balances the head at one arm angle and leaves sin(Δ/2) of its moment over a range Δ.
A spring that must push is hung in the wrong place
Balancing a lamp with a tilting head exactly, with springs of no free length, sometimes asks one spring for a negative stiffness, a spring that would have to push. It looked like a property of the lamp: a light upper arm under a heavy head might admit no design in which every spring pulls. It is a property of one attachment point. The upper arm's spring pulls exactly when the head spring meets the head beyond a point that is always short of the head's centre of mass, 77% of the way on this lamp, and at most 99% even with a head fifty times the arm's mass. Hang the head spring at its centre of mass and every spring pulls on every lamp.
The lamp needs three springs
A desk lamp with a tilting head, balanced exactly with springs of no free length, was built with four springs whose peak pulls add to 162 N. Searched over every attachment point inside the same outline, the least total is 63.4 N, and it is reached with three: the head spring hung at the threshold where the upper arm's spring stops being needed, 77% of the way to the head's centre of mass. Beyond the threshold a second spring is the first one split in two, and splitting a spring never makes it smaller. A design with no parallelogram and six springs is never close: 213 N at its best.
A free length is absorbed where the springs attach
Give every spring of the smallest balanced desk lamp 2 cm of free length and refit the stiffnesses, and the lamp is left with 0.093 N·m of unbalance, growing in exact proportion to the free length. The spare fourth spring that ideal springs had no use for recovers under three per cent of it, and moving the head spring beyond the threshold only makes it worse. What absorbs a free length is a spring whose length barely changes. Re-chosen attachment points buy back a third of the unbalance that way, and pay for it with four times the pull.
The first pull goes to the spring the lamp did without
Give every spring of the balanced desk lamp 2 cm of free length and let it have more pull, and the worst moment it is left with falls from 0.086 N·m at 62 N to 0.052 at 227 N, the most any attachment points inside the outline can do. Half of that improvement costs the first 18 newtons; the second half costs the next 147. Spent on stiffness alone, with the attachment points held, extra pull buys nothing at all. And the first place it goes is the upper arm's spring, the spare that ideal springs had no use for, hung lower on its bracket while the head spring moves out past the threshold. The wrist spring, which carried almost all of the earlier re-attached design's cost, takes the largest share only after 140 N.
One path to the tool
Every mechanism here so far has been a loop, and a loop is why a configuration on this site is a solve. An arm has no loop: its pose is a product of six transforms with nothing to converge and nothing to refuse. The difficulty does not go away — it moves to the other end, where one tool pose has eight answers and a straight line can cost more than the machine has.
The chain that does not close
Every mechanism in these essays so far has been a loop, and a loop is why a configuration here is a solve. An arm has no loop. Its pose is a product of six transforms, evaluated, with nothing to converge and nothing to refuse — and the difficulty does not disappear, it moves to the other end of the problem.
Four numbers or a screw
An arm can be written down as four numbers a joint or as a line in space with a pitch on it. Both are minimal, both describe the same machine to the last bit, and one of them jumps by three hundred and fifty thousand when an axis is tilted by a millionth of a radian.
Where the hand can go
A robot is sold on its reach, which is one number and describes a sphere the arm touches at one posture. The set the tool can actually be put in is an annulus with a hole; the set it can be put in at every orientation is a quarter of that; and reordering the same three links leaves the first unchanged and destroys the second.
Two routes to a Jacobian
An open chain's Jacobian is a list of its joint axes, drawn as lines in the same picture as the arm. A finite difference of its own forward kinematics is a completely different computation and has to agree — and when the two disagreed by 5 × 10⁻⁵, the fault was in a long-standing function that every spatial loop in these essays had been using.
Eight ways to hold the same tool
A six-joint arm asked to put its tool at one place, held one way, has eight answers. Not approximately eight and not eight found by looking — two for the base, two for the elbow, two for the wrist, each exact to a hundredth of a femtometre, and a search from six hundred starting postures finds those eight and no ninth.
The wrist is three joints and one point
Three axes meeting at a point is what makes a six-joint arm's inverse problem solvable in closed form, and it is why every industrial arm is built that way. Move one of those axes by ten millimetres and the construction goes on returning eight confident answers, every one of them out by three and a half.
Where the arm loses a direction
An arm has three singularities and they are three different events. Two of them are inside an ordinary working volume, all three drop the rank by exactly one, and what each one takes away is a screw — a pure force along the arm at the elbow, a pure force across it at the shoulder, and at the wrist a screw of pitch −0.629 that is a force and a couple together.
A straight line at constant speed
Run every joint from its start value to its end value and the tool bows 402 mm off the line between them. Insist on the line instead and the arm will follow it — until the path passes near a singularity, where the joint rates a metre a second demands grow as one over the distance, measured at an exponent of −1.010.
The distance between two poses
Ask an arm to take the shortest route between two poses and three reasonable definitions give three different paths, of lengths 1.223, 1.443 and 1.550 metres. The disagreement is not numerical. There is no distance between two rigid poses until somebody chooses a length to measure a radian in, and on this arm the choice changes which of two poses is nearer at 74.5 millimetres.
The freedom that does nothing
Give an arm a seventh joint and one tool pose stops having eight answers and starts having infinitely many. The arm changes shape while the tool stands still — measured at 3.3 × 10⁻¹² of a metre over forty-one postures — and the elbow runs on a circle that two entirely different computations agree about to a tenth of a picometre.
Pin the tool and it is a loop
Hold an arm's tool still and the open chain becomes a closed one, which these essays have known how to count since their first field. Kutzbach's criterion says a pinned six-joint arm is a structure. At each of its three singularities the measurement says it can still move — the site's founding finding, arrived at from the far end of its own subject.
Where an error at the shoulder ends up
The same angular error at every joint of an arm, and the tool is out by 0.156 mm because of the shoulder, 0.017 mm because of the wrist roll and exactly nothing because of the last joint. The numbers are not properties of the joints. Each one is the distance from the tool to that joint's axis, measurable off the drawing with a ruler.
What a calibration cannot see
A six-joint arm's model has thirty-six parameters and a measurement can find thirty of them. The other six are not hard to measure — they are combinations that move the tool by exactly nothing, at every posture, and no instrument ever built will separate them. The count is 4R + 2P + 6, and it comes out of a rank on four different arms.
The space of configurations
A two-joint arm's configurations are a torus, and drawing one on a page turns it into a square whose opposite edges are secretly the same line. Count the free space on the square and get three pieces; count it on the torus and get one. Eighty single-obstacle arrangements were tried and not one of them cut the torus in two — what does that is a pair of hard stops.
Branches were components all along
Seven words have been used for one thing. An assembly branch, a circuit, an assembly mode, a working mode, a posture and a branch defect are all statements about the connected components of a mechanism's configuration space — and once that is said, a four-bar's two circles, a platform's six modes and a synthesis defect stop being three subjects.
An arm is a tree
The serial field's chains are the ones that never close, and as graphs they are trees. There are 106 distinct arrangements of ten links joined that way, and exactly one of them is the straight arm every essay in the field has drawn — the other 105 branch.
The arm that is a group
A SCARA arm has four joints and a six-axis robot has six, and the usual explanation is that four is enough for the job. The better one is that the job is a four-dimensional group of displacements which is not the symmetry group of any surface — so it cannot be one joint, and four is what it costs. The arm's tool face is level everywhere it can reach, and the reason is not that anybody checked.
An arm's parameters and its poses
A three-link planar arm has three lengths and a tool position that carries a length, so nothing about it is invisible to a measurement — and it is nevertheless the mechanism in this collection where a calibration is hardest, because its parameter count is high, its poses are three-dimensional and its Jacobian is singular where a designer likes to work.
The arm that hits itself
A three-link arm's joint space is a cube of angles and it may not use a fifth of it, because its own material is in the way. The forbidden set does not depend on where the arm is pointing — which is why it can be drawn as a picture rather than described as a volume.
A straight move ends at a stop
A six-joint arm asked to move its tool in a straight line at a fixed orientation keeps the posture it starts in, because changing posture means passing a singularity. Followed from three hundred random poses inside the arm's joint stops, in random directions, the starting posture carries the tool a median of 19.5 cm. Half the moves end at a joint stop, two in five where the arm stretches straight and its posture runs out, and three in a hundred near the wrist singularity, the one the literature on straight-line moves is about. The best of the arm's other usable postures would have gone a median of 34.5 cm, and in a third of the moves it would have gone further. On the way, an earlier count of how many postures an arm can hold was found to be reading its base angle a turn out.
A seventh joint cannot move the elbow
A seven-joint arm can change its shape while its tool stands still, and on a straight move that freedom can be spent keeping its joints off their stops. Spent that way, it lengthens the median move from 23.3 cm to 33.3 cm, against the same arm with its extra joint locked. But the gain is all or nothing. Half the moves a stop used to end gain under a tenth, and a quarter gain more than two and a half times. One stop it never gets round: the elbow's, whose angle is set by the distance from shoulder to wrist and is the same, to 10⁻¹⁵ rad, whatever the other joints do. Nor does it ever reach past the edge of the arm's reach.
Motion that stops
Every other mechanism here moves whenever its input does. A ratchet, an indexer and an escapement are still for most of a turn and moving for the rest, and what decides which is not an equation running out of answers but a tooth arriving at a face. Whether one holds is the sign of a lever arm; what it wastes is two thirds of its own travel.
The mechanism that waits
Every mechanism in this collection so far moves whenever its input moves. A ratchet, an indexer and an escapement do not: they are still for most of a turn and moving for the rest, and what decides which is not an equation running out of answers but a tooth arriving at a face.
A joint that works one way
A pawl either holds a ratchet or is levered out of it, and which one happens is decided by two lines through the contact. One of them is the rule a workshop quotes. The other is the boundary that rule leaves out, and a check written to confirm the quoted rule turned out to be incapable of failing.
The resolution is the pitch
A ratchet's step and a ratchet's error are the same number. Nothing about how well it is made improves that, more pawls divide it by a whole number, and the obvious remedy — cut more teeth — runs into a wall that is geometric rather than practical: at a tooth depth of 0.16 radii the construction stops at twenty-nine.
The arc that is concentric with the pivot
A rotation carries a curve into itself exactly when the curve is an arc of a circle about the centre of rotation. Every exact dwell in a machine is that one sentence applied — a cam's dwell, a Geneva's locking disc, a deadbeat escapement's locking face — and the six-bar dwell that is merely very good is what happens when the curve is nearly one.
The gear with its teeth cut away
Leave teeth on part of a gear's circumference and take the rest off, and the output turns for part of the input's revolution and stops for the rest. It is the cheapest intermittent drive there is and it engages at full speed, so its output's velocity has a step and its acceleration is not a large number — it is not a number, and the way to report that is to watch a difference quotient refuse to converge.
Where the tooth lets go
A pair of pallets spanning a whole number of tooth pitches and a half advances the escape wheel exactly half a pitch every beat, and that half pitch divides into the impulse and the drop with nothing left over. Drop is not chosen. It is whatever the impulse leaves, and on a thirty-tooth wheel it is two thirds.
The wheel that goes backwards
While a pendulum finishes its swing the escape wheel is doing something, and what it does is decided entirely by the shape of the face the tooth is resting on. An arc about the pallet arbor sends it nowhere — not nearly nowhere, the same double at every sample. A flat cut tangent to that arc is dead at exactly one point of itself, and it is the one point the tooth never rests on.
The angle that holds the lock
A locking face cut exactly concentric with the pallet arbor has no tendency to hold itself: the tooth's push aims straight at the pivot and its moment is zero, so the smallest disturbance opens the lock. Tilt the face and the moment becomes ρ sin δ — and the same tilt, by virtual work, is exactly the rate at which the wheel is driven backwards. Draw and recoil are one angle read twice.
Detached, and safe while detached
A lever escapement touches its balance for a twelfth of each beat and leaves it alone for the rest. What connects the two is a pin entering a radial slot — the same pair a film projector's Geneva drive is made of, solved by the same eight lines — and the sine rule then fixes the balance's lift angle at 44.4°, which is the number a Swiss lever is specified at.
One test, three mechanisms
Whether a pawl holds, whether an escapement's lock draws itself deeper, and how much a four-bar's coupler can do for its rocker are the same question asked three times: on which side of a pivot does a contact normal pass? All three are one cross product, none evaluates a force, and the three answers are used for completely different things.
Where the input stops deciding
Give a four-bar its crank angle and its coupler is somewhere definite. Give an escapement its pallet angle and the wheel may be in any of three places, because what settles it is which face of which pallet a tooth is against. The path crosses itself and no amount of solving removes the crossing — the state of these mechanisms has a discrete part, and that is what makes them a different kind of object.
One piece, and still not reachable
The site decides whether two configurations can be joined by asking whether they are in the same connected component, and that relation is symmetric because a path run backwards is a path. A one-way mechanism breaks the symmetry and leaves the connectivity alone: its free space is a single interval with no barrier anywhere in it, and about half of the ordered pairs of states cannot be joined by any admissible motion.
Two pins and no dwell at all
Put a second pin on a Geneva's crank and the wheel indexes twice a turn instead of once, at a quarter of the acceleration for the same output rate. Put a third on a six-slot wheel and it never rests; put a third on an eight-slot wheel and two pins meet in two slots and it jams. Both of those look like separate conditions and are one, and the boundary between them is an equation in integers with exactly three solutions.
A ratchet with no teeth
A roller clutch holds one way and runs free the other with nothing on it a tooth could be called. Whether it grips is a question about friction, which no drawing settles. How far it moves before it does is geometry: a clearance divided by the gap's own slope, so its lost motion is a length rather than a fraction of a pitch — 1.4° against a 24-tooth ratchet's 15° — and unlike a ratchet's it is not divided by adding more holding elements.
When the index law becomes a choice
A Geneva's motion law is forced by its slot count and a cam indexer's is chosen, so the fair comparison gives the cam the Geneva's own index angle and step. On peak acceleration the cam wins only below a slot count that depends on the law — 5.19 for cycloidal, 6.23 for modified sine, 8.06 for simple harmonic — and above it the Geneva does. What no slot count removes is the step: the pin arrives with an acceleration of exactly tan(π/n).
The disc decides the pin count
A Geneva's pin count is usually bounded by the slots: p indexes must not overlap, so fewer than 2n/(n−2) pins fit. The other half of the mechanism has its own inequality and nobody had measured it. The locking disc must be cut away wherever the wheel passes through it, that cut-away is wider than the index sweep at every slot count, and it is the binding condition everywhere — one pin only, from four slots upward.
A flat holds the wheel to second order
A Geneva wheel is held between indexes by a disc on its driver sitting in a concave arc, and that disc is so large that one pin is all a wheel of four slots or more can carry. Hold the wheel on a flat chord instead and the disc must shrink to the crank times sin(π/n). Its cut-away falls from 236° to 127° on six slots, and a second pin fits at every count from four to twelve. The cost is the hold itself. An arc lets the locked wheel rock in proportion to the running clearance and a flat as its square root: 0.022° against 1.48° at a clearance of 0.02 on a crank of 30.
A V holds the wheel to first order
A Geneva wheel locked on concave arcs holds to first order and carries one pin; locked on flat chords it carries two and holds only to second order. Cut the lock as a shallow V instead — two faces tilted a few degrees off the chord — and it keeps both. Its disc is the crank times sin(β + γ), small enough for a second pin up to 34° at four slots and 5.6° at twelve. It rocks as acos(cos γ − c/a) − γ, which is linear in the clearance below 2a sin²γ, and at a running clearance of 0.1 on a crank of 30 it holds five to eight times tighter than the flat. The trade was between two shapes, not a law.
A curved slot moves the step
A Geneva's pin arrives with an acceleration of tan(π/n) of the crank's speed squared, and no choice of entry direction removes it. A choice of slot shape can: bend the slot's mouth with the crank's own circle and the slot absorbs the pin's acceleration instead of handing it to the wheel. But the pin leaves by the mouth it came in by, and there the same bend adds to the step. Solved at four slot counts and seven curvatures, the entry and exit steps are tan(π/n)(1 ∓ κ̂) and always sum to 2 tan(π/n). A curved slot moves the step from one end of the index to the other, every bend raises the peak, and no slot tried beat the straight one.
A steered pin keeps only the window
A Geneva's step at entry and exit belongs to its pin's circle, not to its slots. Carry the pin in a radial guide on the driver and set its radius with a stationary cam, and the wheel can be given any law at all — the pin is simply where the driver's ray meets the slot's line. Cut for a modified trapezoid, a six-slot wheel starts and stops with no step and peaks 14% below the crank pin, while the cam moves the pin by 1.2% of the crank. What the circle cannot hand over is its window: no ray before the crank pin's entry meets a resting slot, so the index can never be longer, and every degree spent holding the wheel in its slot comes out of it as the square.
A wheel held by its slots needs three pins
A steered pin can hold a Geneva's wheel by running along a resting slot, but every pin is in some slot for exactly the crank pin's window, 180° less 360°/n of every turn, because it can leave a slot only by the mouth it will reach. Two windows of less than half a turn can never cover a turn, so two pins cannot hold a wheel without its disc at any slot count. Three can, on six slots and more, and four on four. At the Geneva's own share of index to dwell the index is no harder than the crank pin's, and the slots hold the wheel tighter than any lock in the field. The disc's work moves to the steering cam: on a run the cam's pressure angle is the run's own angle, and the six-slot drive's runs are 40°.
More than one input
Every mechanism here so far has had one input, and its output has been a function of it. A differential does not: its cage turns at the mean of two wheels, and neither of them decides anything alone. What a gear train has is not a ratio but a relation — a plane of permitted motions — and every number a gearbox is sold with is that plane cut by a brake, a clutch or a choice of which shaft is driven.
Two inputs and one output
Every mechanism in this collection so far has had one input, and its output has been a function of that input. A differential does not. Its cage turns at the mean of two wheels, so knowing one of them tells you nothing at all about where the third shaft is going — and that is not a complication of the mechanism, it is a different kind of object.
A ratio is a null space
Write a gear train as a graph — bodies for vertices, meshes for edges, and on every edge the body the two axes are stationary in — and one formula covers a countershaft gearbox, a planetary, a harmonic drive and a car's differential. The ratio is the null space of a matrix whose entries are tooth counts, so it comes out as a fraction and not as a number that is nearly one.
The lever that is the gearset
The lever diagram of an epicyclic is usually offered as a mnemonic. It is exact, and the reason is a fact about the null space: a gearset whose frame carries no teeth can turn as a block, and that one motion supplies the coordinate every member is plotted at. Where the line crosses the axis is the member standing still, and the ordering of the members on the lever settles which gears are reductions and which run backwards, without a formula anywhere.
Holding a member chooses the ratio
A gearset offers a plane of motions and a shift element is one linear condition, so a gear is a line in that plane. Enumerate every brake and every clutch and a Ravigneaux's twenty-seven combinations collapse to seven ratios — eighteen of them being the same gear, because locking any two members at all locks the whole gearset solid. Neutral is not one of the seven, because neutral is not a gear.
Four speeds from two numbers
A Ravigneaux gearset has five tooth counts and gives seven exact ratios. Two of the counts do not appear in any of them — the short planet is an idler and its size is free — and the remaining three enter only through two dimensionless numbers, so the whole set of shift ratios of a four-speed automatic is a function of ring-over-sun and ring-over-the-other-sun. A Simpson three-speed is a function of one number.
The steps are not free
A gearbox is supposed to have equal steps between its gears, so that the engine returns to the same speed after every shift. A gearset has one or two numbers to spend on three or four gears, so from the third one the steps are a consequence rather than a choice — and asking for them to be equal turns out to be a quadratic whose root is the golden ratio, realised in tooth counts by consecutive Fibonacci numbers.
One wheel on ice
A differential with one wheel stopped turns the other at exactly twice the cage, and the relation it imposes is satisfied the whole time — nothing has failed, nothing is confused, and the reason the car does not move is not in the kinematics. What is here is the other half: a locked axle is an overconstrained mechanism, and the sliding it produces is 2π times the track per circle driven, whatever the radius.
A ratio with no steps in it
Push a variable pulley's sheaves together and the belt rides further out. The other pulley's radius is then not a choice — the belt has a fixed length — so it is the root of an equation, solved rather than set. The rule of thumb that says the two radii add to a constant is true to first order and wrong by 7.4% of the ratio at full shift, and the departure has a closed form.
A hundred to one from a difference of one
A harmonic drive reduces by a hundred to one in a single stage with two gears in it, and the hundred is the flexspline's tooth count divided by the two teeth the circular spline has more than it. The same null space that answers a planetary answers it. What each of the three single-stage reductions pays for that arithmetic is different, and the compound epicyclic's price is a pair of meshes whose centre distances differ by half a tooth.
Three mechanisms, one subtraction
A micrometer's differential screw, a chain hoist's differential pulley and a robot joint's compound epicyclic look nothing like each other and are the same device. Each takes two nearly equal quantities and returns their difference, each buys its enormous ratio with that difference, and each carries the same conditioning number — |a/(a−b)| — measured here by perturbing the mechanisms rather than by quoting the formula.
The reductions a planetary cannot give
One epicyclic offers six ratios, and the formula for each of them suggests the whole positive line is available. Sweep every design that can actually be cut and assembled and the reachable set has a hole in it running from 1.630 to 2.586 — the width of which has a closed form — and a reduction of exactly 2, the most ordinary thing anybody asks a gearbox for, sits in the middle of it.
The transmission angle has no size
The geometric half of force transmission is an angle in a triangle whose three sides scale together, so it is the same at every size — 54.31° at its worst on this machine, whatever units the drawing is in. Which means a measurement that cannot recover a single length recovers the whole of what this field computes.
Two shafts that must be in line
Asking a two-stage gear train for a ratio is easy. Asking it for a ratio and for its input and output shafts to be coaxial is asking for a solution of two equations in four integers, and there is no reason for one to exist. A twelve-to-one reverted train needs a sixty-three-tooth wheel before it has any solution at all — while sixteen to one, a larger ratio, manages with fifty-six.
A bounded ratio made unbounded
Split an engine between a variator and a straight path and add the two with a planetary, and a variator that spans 3.76 to one becomes a machine whose ratio passes through infinity. The setting at which the output stands still is the planetary's own tooth ratio and nothing the belt does moves it. The price is a sensitivity that grows as the reciprocal square of the distance to that setting, and the variator's one and a half per cent becomes a hundred and ten before the setting is reached.
Sliding the travel across the pole
A power split's ratio has a pole the variator's own tolerance makes unusable, so the question is where to put the variator's travel relative to it. With a tolerance that is one number, the best forward span comes where the travel's top just meets the trim — 1 + τ(1 − r)/p, with no gearset in it. With the variator's real tolerance, which grows along its travel, that peak flattens into a plateau: the pole can be moved well inside the travel, buying reverse, for under a tenth of the forward span.
The power that goes round twice
Sliding a power split's travel towards its pole buys ratio span for nothing, on the kinematics. It is not for nothing. The variator's own branch carries v/(K−v) of the engine's power and overtakes it at exactly half the way to the pole, and the tolerance trim the span was computed from is not reached until the variator is rated for six times the engine — which no machine is.
Every power split is on one curve
There are twelve ways to attach an engine, a variator and an output to a planetary's three shafts, and asking which is best turns out to be the wrong question. On 1,994 splits across all twelve and nine gearsets, the output's span is (1 + R·a)/(1 + a) to 5 × 10⁻¹⁵, where R is the variator's own span and a is the power the variator carries over the power the gears carry. Every split lies on one of three curves of span against the variator's largest share. The previous essay's 1.70 is 2 − 1/R on the worst of them, and a split beats its variator's own span only by making the variator carry more than the engine.
A second planetary lets the variator carry a difference
On a single planetary every power split lies on one of three curves of span against the variator's share of the engine's power. Build the split on a four-shaft Simpson or Ravigneaux instead and nothing changes while the fourth shaft is free: 3,588 splits, all on the curves. The one new arrangement is the variator between two of the gearset's shafts. That compound split is an input-coupled split and an output-coupled split in series, and its variator carries only the difference of their shares — so at every share below one it can span more than any single planetary, the square of (√R + k)/(1 + k√R). It needs a lever with the variator's shafts exactly its share apart, which real gearsets do not have beyond a share of about a third.
One cross-ratio decides the compound split
A compound power split, its variator between two shafts of a four-shaft gearset, can beat every split on a single planetary, but four real gearsets let it do so only up to a variator share of about a third. Whether teeth can do better turns on one number: the cross-ratio of the gearset's four shaft positions on its lever. A third needs 2.52 and a half 4.67. Every Simpson inside a ring of 100 teeth stops at 3.87, a share of 0.454. A Ravigneaux with large suns and 13-tooth planets reaches 5.61, a share of 0.542, and solved on its lever it passes a half. With planets of 17 teeth, neither family does.
The compound split has gearsets of its own
A compound power split depends on its gearset through one number, the cross-ratio of the four shafts on its lever, and a variator share of a half needs 4.67. Simpsons stop at 3.87 inside a ring of 100 and Ravigneaux reach it only with 13-tooth planets. Two simple planetaries can be joined into four shafts in eighteen ways, twelve of them distinct gearsets, and ten of the twelve reach a half with ordinary ring-to-sun ratios. The cross-ratio is then no longer the question. Three joins reach any cross-ratio by setting two shafts nearly together and pay with 2.7 to 4.2 times the engine's power inside, and one, carrier to ring both ways, never reaches a half. A sun-to-carrier, carrier-to-ring join builds the lever with planetaries of 26/52 and 41/71, and nothing inside it carries more than 0.80 of the engine's power.
The winning join is a gearbox too
The sun–carrier, carrier–ring join of two simple planetaries builds a compound power split's half-share lever with ordinary teeth. Locked out of the split and given clutches and brakes, the same four shafts give five forward ratios below the split's range, from 2.73 to direct drive, and an overdrive of 3 above it. The two gears that brake a variator shaft sit 1.76 and 1.74 outside the split's ends, and three gearsets with the same cross-ratio put them at the same steps. Its planets space evenly, three and four to a set, and its meshes lose 0.41% of the engine's power along the split where the Ravigneaux that builds the same lever loses 1.01%. What none of these gearsets gives, with the output where the split needs it, is a usable reverse.
The motion, not the mechanism
Everything else here is about a machine. This is about the motion a machine makes, near one instant — where the moving plane is turning, which of its points are going straight, which of them could be replaced by a single pivot and for how long. Two linkages with nothing in common that agree on those numbers are interchangeable, and the mechanism turns out to be one of infinitely many ways of producing a motion rather than the thing the subject is about.
The mechanism drops out
Every other field here is about a machine. This one is about the motion a machine makes — a plane sliding over a plane — and near any instant that motion is a handful of numbers with no linkage in them. Two mechanisms that agree on those numbers make the same motion, and one of them can always be thrown away.
Every point has a centre
A point of a moving plane traces a curve, and near an instant that curve is a circle. Which circle is decided by one relation with two numbers in it — the same relation for every point of the plane at once, and it was written down in 1830 with no derivatives visible anywhere in it.
The circle of points going straight
At any instant some points of a moving plane are travelling in a straight line. They form a circle — not nearly a circle, a circle — and the reason is one real coefficient in a quadratic. A coupler curve has an inflection exactly when that circle sweeps over the tracing point, which turns out to be rare.
The other pole
The instantaneous centre is the point of a moving plane that is not moving. There is a second point that is not accelerating, it is somewhere else entirely, and over a full turn of one four-bar the two are never closer than one and a half coupler lengths and get as far apart as twenty-two.
A construction with no arithmetic in it
Everything in this field so far has needed the motion differentiated. Bobillier's theorem gets the pole tangent — the one quantity that otherwise needs a derivative — out of two lines that are already drawn on the mechanism, and the inflection circle follows from three points and a pair of compasses.
Where the curvature stands still
One derivative further on there is a second locus: the points whose path curvature has momentarily stopped changing. The expression whose zero set it is looks like a quartic, its fourth-degree terms cancel, and the cancellation is measured at ten to the minus fourteen rather than assumed.
The straightest point there is
Where the circle of points going straight meets the cubic of points whose curvature is standing still, there is one point whose path is straight and staying straight. Neither Watt nor Chebyshev put their tracing point there, and moving Watt's to it more than halves the error over the whole stroke.
Four positions brought together
Burmester's construction finds the points of a moving plane whose four prescribed positions lie on a circle. Bring the four positions together and the curve it draws has to become the cubic of stationary curvature — and the gap between the two closes as the square of the spread, with the exponent measured rather than assumed.
Where a curve has a corner
Some coupler curves have corners in them — points where the curve stops, turns round and comes back. A corner happens where the tracing point is momentarily still, the only point of a moving plane that is momentarily still is the pole, and so the corners of a coupler curve are decided entirely inside the coupler's own plane.
Exact because two circles roll
Watt's straight line is straight to nine parts in a hundred and Chebyshev's to twelve. Here is one that is straight to nothing at all — no error term, no working range, no approximation anywhere — and the reason is that its two centrodes are circles, one rolling inside the other at exactly half its radius.
The circle a point stays on longest
One condition further on are the points whose path holds a circle to fifth order. A planar motion has at most four; two of them are always the mechanism's own pins, and the other two are real for two thirds of a turn and complex for the rest — a count that changes while nothing about the mechanism does.
A curvature is a size with a minus sign
Everything in the curvature field is a similarity invariant in shape and a reciprocal length in value. Scale a moving plane and its inflection circle scales, its cubic of stationary curvature scales, and every curvature it computes is divided by the factor — so a bigger machine traces gentler paths than its drawing suggests.
How long a pivot stands in for a linkage
Throw the four-bar away and replace a coupler point by a crank pivoting at the centre of its path's curvature. The two separate like the cube of the crank step for an ordinary point, the fourth power on the cubic, and the fifth at a Burmester point — three exponents that are the whole field, measured with no derivative anywhere in the measurement.
The linkage, put back from two curves
This field opened by saying the mechanism drops out, and that every planar motion is one curve rolling on another. Both are true and neither had been measured. The rolling reproduces the four-bar's own placement to a residual that quarters when the sampling halves, and the two curves lay equal arc to a part in a billion.
The two numbers are the curves' own
Every essay in this field runs on the inflection circle and the pole tangent, and every one has taken them from the motion's derivatives. They are the two centrodes' radii of curvature at the pole, combined harmonically — measurable off the curves with no mechanism in the arithmetic, and carrying a sign that changes eight times in a turn.
The frame seen from the coupler
Hold a four-bar's coupler still and let its frame move, and every construction of the curvature field has a counterpart. The frame's points that travel straight lie on the ordinary inflection circle reflected through the pole — found by differentiating the swap and, independently, by solving the coupler-held four-bar, both to 10⁻¹⁴ — and a point and the centre of curvature of its path trade places exactly. The inverse motion of one four-bar is the ordinary motion of another.
The other pole is on the circle
The point of a moving plane with no acceleration depends on how fast the input is driven, which is why it was set apart from the rest of this field as the one object that is not geometry. It is not free, though. It always lies on the inflection circle — the geometric locus of points going straight — at an angle from the circle's diameter whose tangent is the plane's angular acceleration over its angular speed squared. A drive chooses one angle, and the geometry has already chosen everything else.
The third pole goes anywhere but the circle
The velocity pole is fixed by the geometry and the acceleration pole by the geometry up to one number, its place on the inflection circle. The next one, the point of a moving plane whose jerk is nought, depends on two numbers of the drive — and two numbers are enough to put it anywhere. Every point of the plane off the inflection circle is the third-order pole of exactly one drive, because a body point's jerk is affine in the crank's acceleration and jerk. On the circle only two points qualify: the velocity pole, and Ball's point, where the path is straight to third order.
No drive holds the third pole
At one crank angle a drive's acceleration and jerk can put the coupler's third-order pole on almost any point, making that point's jerk nought. Over a stretch of the stroke they cannot. A crank's schedule is one function, its speed against its angle, and its jerk number is fixed by its acceleration number: λ = κ′ + 2κ². So the two numbers a point needs at every angle must satisfy that relation, and on the standard four-bar they miss it by as much as the numbers themselves, agreeing only at isolated angles. The best schedule a crank can follow takes a fifth to a third off a coupler point's jerk, and no more.
The smoothest coupler point is chosen in time
A crank's speed can be scheduled to cut a coupler point's jerk, though never to nought. Which point should a designer free to choose it pick? Measured as the schedule essays measured jerk, per crank speed cubed, the answer is a point whose schedule swings the crank's speed four to one, and it is a cheat: that measure divides by the speed it changes. Measured in time over a stroke of fixed duration, per unit of the point's own travel, the smoothest point of the standard four-bar is (−0.02, −0.80) in the coupler's frame. Its schedule cuts its jerk by 72% with a speed swing under two to one, and it is 63% smoother than the best a steady crank finds anywhere. Ball's point, the instant's straightest, lies in a second, shallower valley four coupler lengths away.
A crank that must come round picks another point
A crank scheduled over one stroke can cut the smoothest coupler point's jerk by 72%. A crank running continuously must come back to the speed it started at, and its working stroke must keep its share of the cycle. Held to both, the same point keeps two-thirds of its gain only by putting five and a half times a steady crank's jerk into the return, and no schedule beats a steady crank on both strokes. Choose the point for the whole turn instead and a gentle schedule, its speed swinging only 1.66 to one, cuts the working stroke's jerk by 74% and the return's by 18% together. That point is three coupler lengths from the first.
Wheels, and where they may not go
A rolling wheel forbids a velocity, not a position. It may not go sideways and it can still be parked anywhere, at any angle — and the gap between those two sentences is worth an exponent, a shortest path that is a whole circle, and a ball that comes back turned by the area it went round.
A constraint that takes nothing away
A rolling wheel forbids one direction of motion and removes no coordinate from the mechanism's description. It cannot slide sideways and it can still be brought to any position at any heading — and the gap between those two sentences is the whole of this field, because in every mechanism built of pins and slides the two agree.
One character apart
Two mechanisms with three coordinates, one constraint row of the same shape and two controls each. In one of them the angle in the row is a coordinate; in the other it is a constant. The first can be driven anywhere and the second can never leave a line, and Frobenius' theorem decides which is which without integrating anything.
The motion left over by going nowhere
Drive forward, turn, drive back the same distance, turn back the same angle. Every leg is undone by another leg and the mechanism does not come home — it has moved sideways, in the one direction it is forbidden to move in. The leftover has a name, a formula, and a measured exponent of 1.997.
How many wiggles
A bracket of two permitted directions may point somewhere new; the bracket of that with a permitted direction may point somewhere newer still. How deep the process goes before it stops is an integer — 2·3 for a wheel, 2·3·4 for a car, 2·3·4·5 for a car and trailer — and the same integer turns up as the exponent of a manoeuvre nobody told it about.
Parking is an exponent
Four legs — forward on left lock, forward on right lock, back on left lock, back on right lock — return a car to its own heading and to its own place along the road, exactly, and move it sideways by 4R sin φ tan(φ/2). Halve the room and the gain quarters, so the number of shuffles goes up by four and the distance driven doubles.
The path a towed wheel takes
A towed axle obeys one line: roll along your own heading, and stay attached. Nothing tells it to keep its distance from the hitch and it keeps it to 10⁻¹³ anyway, it settles onto a circle of exactly √(R² − L²), and the residual against that is not the integrator — it is the difference between a circle and the polygon it was sampled as, and it falls by four when the sampling doubles.
Which way did the bicycle go
Two tyre tracks in mud, and a question with a definite answer. The rear wheel is towed, so its tangent extended forward by the wheelbase must land on the front wheel's track — and it does, to 0.13 mm one way round and 447 mm the other. The test needs neither the wheelbase nor the direction of travel, and it returns both.
A circle for the first millimetre
The shortest path for a car that may not reverse, from here to a point one millimetre to the side at the same heading, is 31.417 m for a five-metre turning radius. The shortest path to a point twenty metres to the side is 31.416 m. The cost of going sideways is not monotonic in how far sideways, and below a crossover at 2.956 R it is exactly 2πR + δ.
The ball that remembers where it has been
Roll a ball round a closed loop on a table without ever twisting it, and it comes back to the same place pointing somewhere else. The angle is the loop's area divided by the square of the radius — 0.0016 radians for a 2 mm square under a 50 mm ball — and it is exact in the limit with a departure that is second order in the angle itself.
The wheel that forbids nothing
Every wheel contributes the same row to the same matrix, and whether that row is a constraint on the vehicle or a statement about the wheel's own speed is decided by one factor of sin γ. At γ = 0 the vehicle may not move across the wheel; at 45° the row says nothing about the vehicle at all, and sideways costs exactly what forwards costs — to the last digit, and at no other angle.
Every axis through one point
Bolt several rolling wheels to one rigid body and they impose one condition between them: every axle line must pass through a single point. The familiar steering formula falls out of it as a consequence rather than being quoted — cot δₒ − cot δᵢ = 0.574074 at a turn of six metres, of eight, of twelve and of twenty, on a track of 1.55 m and a wheelbase of 2.7.
The angle that doubles
A trailer a hundredth of a radian out of line decays back into line as e^(−s/d) driving forwards and grows as e^(+s/d) reversing — doubling every 4.16 m for a six-metre trailer. And a jackknifed rig is not a rig that has lost anything: its growth vector is 2·3·4·5 at a hitch angle of zero, of ninety degrees and of a hundred and eighty.
A wheel that cannot report its radius
Rolling relates a wheel's turning to a vehicle's travelling, and the relation has a length in it. So a rolling constraint is the one place in these essays where an angle measurement does carry a size — and the size it carries is the one thing a vehicle's own odometry can never separate from its wheelbase.
The road a wheel carries with it
A taut strand on a pulley is a rolling contact: the material at the tangency is at rest against the surface, and the ratio between two bodies on one span is the ratio of their arms. But this rolling constraint integrates, where a wheel's does not — and the difference is that a strand rolls along a line and a wheel rolls across a plane.
One bracket, two subjects
A wheel can be parked sideways because the forbidden direction is the bracket of two permitted ones. A mechanism's motion is a group when the brackets of its permitted twists are already permitted. Same operation, same two plateaux, and the two fields want opposite answers — which is why the two are named for different things here, with each one saying beside itself that the other exists.
A bearing is a planetary with no teeth
Roll a ball between two races and nothing but the two rolling constraints decides how fast its centre goes round. Solved, they put the cage at (1 − d/D)/2 of the inner race's speed — always less than half — and the gear field's train solver, handed a planetary with a sun of D − d teeth and a ring of D + d, returns the same fraction exactly. And a tapered roller rolls without slipping along its whole line only if its axis meets the bearing's at the apex, the wheel field's concurrency one dimension up.
What a wheel counts on a rough road
An odometer counts a wheel's turns and multiplies by its radius, and on a flat road that is the distance travelled. On a rough road it is not, and it is not the road's length either. It is the length of the path the wheel's centre follows — exactly, because the contact is the wheel's instantaneous centre. A wheel smaller than the road's tightest dip rolls over all of it and counts the road's full length; a larger one bridges the dips, pivots over the crests, and counts less, falling towards the true distance as λ³/24(R + ρ)². On a road 2.4% longer than its wavelength, a 50 mm wheel overcounts by 2.4% and a 2.4 m one by 0.03%.
A second wheel cannot correct the first
A wheel on a rough road counts the length of its centre's path, not the distance travelled. Put two on one rigid frame and average them, and the error does not shrink at any wheelbase: both centres run on the same curve, fixed by the road and the wheel's size, and each wavelength of travel carries each of them once round it. A half-wavelength wheelbase smooths the count's ripple and rocks the frame instead. What does work is two wheels of different sizes. A bridging wheel's error falls as the inverse square of its radius, so two radii measure their own error: wheels of 0.6 and 1.2 m that overcount by 0.339% and 0.099% extrapolate to 0.018%, knowing nothing about the road.
A real road sets its own power
On a road of one sine, a wheel's overcount falls as the inverse square of its radius, and two wheels extrapolate it away. A real road is a spectrum. Over the band road standards describe, every wheel smaller than the road's tightest trough — about a metre and a half on an average road — counts the road's whole length, and its size makes no difference at all. With the texture below that band, bigger wheels do bridge more, but the overcount falls as the two-thirds power of the radius, a power the road's spectrum sets: 2(3 − w)/(5 − w) for a spectrum falling as n^−w. The inverse-square recipe removes a sixth of the error.
Three wheels read the road they measure
A wheel's overcount on a real road falls as a power of its radius that the road's spectrum sets, so two wheels cannot extrapolate it away without being told the power. Three can fit it. Rolled together over 150 m of a class C road textured to 1 cm, wheels of 0.3, 0.6 and 1.2 m read a power of 0.64 against the law's 2/3, read the road's class from its level, and extrapolate the distance to 2 × 10⁻⁵ where the smallest wheel alone is out by 10⁻³. The standard band's floor, which no practical wheel bridges, turns out to lie on the same law and is extrapolated with it. The method is worth five times on a smooth road and five hundred on a rough one, and near the end of the road's texture it reads the end instead.
A load reads the road a radius cannot
Press a tyre harder and its contact patch grows, and a patch averages the road where a rigid wheel rides its peaks. Averaging is linear, so the overcount it leaves falls as the patch's first power and scales with the road's roughness in exact proportion, not as the two-thirds power three wheel sizes read. One 0.3 m tyre at deflections of 3, 9 and 27 mm fits that law and extrapolates the distance to 3 × 10⁻⁷ on a smooth road and 6 × 10⁻⁶ on an average one, against 2 × 10⁻⁵ for three rigid wheels. On the roughest roads its lightest patch lifts off in places, and it does worse than they do.
The shape is the unknown
Everywhere else here a body has a shape and the question is where it goes. Put two bodies on fixed centres, require them to stay in contact, and the second shape stops being a choice: it is the envelope of the first one's positions, and there is exactly one of it. One routine cuts a gear tooth out of a straight edge, a cam out of a roller and a rotary engine's rotor out of its housing.
The second shape is not a choice
Two bodies on fixed centres, told to stay in contact. Give one of them a shape and the other one's shape is no longer available to be designed — it is the envelope of the first one's positions, there is exactly one of it, and one routine computes it for a gear, a cam and a rotary engine alike.
Any shape has a partner
Conjugate action does not pick out the involute. Hand the construction a flank invented on purpose to be nothing in particular and it returns a mate that holds the ratio exactly — so the question a tooth form answers is not whether it can transmit motion, and every real reason for choosing one is somewhere else.
Rolling at one point only
Two wheels in mesh are usually described as rolling. Their pitch circles are — those are centrodes, and centrodes roll — but the surfaces that are actually touching slide against each other everywhere except at one instant, and the sliding is the largest velocity in the mechanism.
The tool is the definition
There is no curve anywhere on the cutter that makes an involute gear. It is a straight edge, dragged past a turning blank, and the involute is what the motion leaves behind — along with a fillet that is a corner's path, a contact locus that comes out a straight line, and a base circle that is measured rather than drawn.
The shape that does not mind where the shafts are
Two tooth forms, both exactly conjugate, both in use for centuries. Move the shafts five hundredths of a millimetre apart and one of them wants a different shape and the other does not — and that single measurement is close to the whole reason every gear cut since about 1900 is an involute.
A ratio that is a function of the angle
A gear pair is usually two circles rolling. Ask instead for an output that runs forty per cent fast for half a turn and forty per cent slow for the other half, and the two shapes that deliver it are not a design decision — the demand fixes both pitch curves completely, and the only question left is whether they close.
The demand that cannot be met
Ask for an output rate and the two pitch curves follow with no design step in between — so the interesting question is not how to draw them but which demands admit any pair of wheels at all. The answer is one equation on a whole function, it is about the demand's mean and nothing else, and five per cent of error leaves the output eighteen degrees out after a turn.
A rotor nobody drew
The rotor of a rotary engine has three corners, three flanks and one job: to stay in contact with a housing while turning at a third of the shaft's speed about a centre that orbits. Given the housing and that motion, the rotor is not designed. It is computed, corners and all, by the routine that cuts a gear tooth.
Three chambers and a constant
A rotor with three corners divides its housing into three chambers, and as the shaft turns they trade area: one grows exactly as fast as the other two shrink. The total does not move — and the fact that it does not is a measurement of whether the corners are actually touching the wall.
Eleven lobes from twelve pins
A ring of round pins, a disc on an eccentric, and a reduction of eleven to one with no gear teeth anywhere. The disc's profile is not designed: one pin generates one lobe of it, the other ten lobes are the same curve, and the count that decides the ratio is a count of lobes on a shape nobody drew.
The cutter takes back the tooth
Undercutting is usually explained as a shape: a tooth with a waist in it. It is better understood as an event — the corner that leaves the fillet comes back through flank the straight edge has already generated — and seen that way the threshold at seventeen teeth is a comparison of two measured points that passes through zero.
Where two shapes stop touching
A conjugate pair is exact at every instant it has a contact. It does not have one for ever: a profile is an arc rather than a curve, and both ends of that arc are somebody's decision — which is why the useful question about a pair of shapes is not whether they mesh but for how long.
Rotors that mesh and cannot drive each other
Two identical lobed rotors on shafts turning one to one are each other's conjugate: give half of a lobe and the meshing equation computes the other half so exactly that the rotor is its own mate. The pair holds its ratio at every instant and still cannot drive itself, because the one contact between them pushes the driven rotor backwards for exactly half of every turn.
The clearance that is the seal
Two rotors that are each other's conjugates touch at every angle of their turn, so cutting both back by the same amount leaves exactly twice it between them, everywhere. Open the shafts by the same amount instead and the gap runs from four per cent of it to all of it. And a pair that is not conjugate has no seal to cut: over one lobe pitch it swings from two and a half units inside itself to two and a third apart.
A twist steadies what it cannot tighten
A helical rotor's sections are at different phases of the same mesh, so the clearance a machine has at one instant is a window along its own profile rather than a point on it. A wrap of exactly one lobe pitch holds the seal's open area constant through the turn — every harmonic at once, whatever the profile — and leaves its average, its tightest place and its widest place exactly where they were.
What a second contact is for
Two identical rotors that are exactly each other's conjugates cannot drive each other, and the reason has nothing to do with conjugacy. A ring of pins and the disc they generate is just as exactly conjugate, has nine or ten contacts at once instead of one, and never loses more than a fifth of the arm its geometry allows.
A clearance leaves one contact
A cycloidal drive's disc, exactly as its pins generate it, touches nine or ten of its twelve pins at once, and that is why it can drive where two identical rotors cannot. Cut the disc two hundredths undersize, as a real drive is cut, and a rigid disc turned until it touches meets one pin, at every angle of the eccentric and in both directions. The arm that pin gives is never worse than the exact disc's best. What a clearance takes away is load sharing, and deflection gives it back: contacts that may give by the whole clearance put six or seven pins within reach.
A disc whose roots are not sockets
Put a cycloidal drive's eccentric at exactly the pin circle's radius over the pin count and every root of its disc is a round socket; take any smaller eccentric and the roots are smooth. The sockets looked like a benefit — a pin in one closes its gap both ways at up to the full pitch offset. Measured from a half of that eccentric to all of it, they are the opposite. The socketed disc touches nine or ten of its twelve pins, every smaller one touches all twelve, and its first contact keeps the smallest share of the pitch offset of the seven. What a smaller eccentric really costs is the pitch offset itself: the free turn per unit of clearance is 0.0365 divided by the eccentric's share.
The flank that trims the root
A cycloidal disc's roots are trimmed, and its pins start standing clear, above an eccentric that was known only for one drive: 0.974 of the pin circle's radius over the pin count, for twelve pins of radius 5 on a ring of 60. The trimming is a bend. Either side of each root the flank of the pin's path curves towards the disc, and where that bend is tighter than the pin the offset folds. The bend has a closed form, so the threshold does too, √(1 − ρ²(N + 1)³/27(N − 1)) for pins of radius ρ, and it falls well below one for fat pins or many of them. On every drive measured, the classical eccentric loses contacts and lever together.
A disc cut at its threshold grows a point
A cycloidal disc's roots stay untrimmed up to an eccentric where the pin path's flank bends with the pin's own radius, and it looked as though a disc cut exactly there would have its most conformal contact, the pin and the flank sharing a curvature. It has its least. The disc's edge is the path offset inwards by the pin's radius, so where the path bends round a centre on the disc's side the edge bulges towards the pin with the path's radius less the pin's: at the threshold, a point. A line contact's pressure per unit load there rises as the inverse square root of the distance to the threshold, and the pins passing it are carrying load. Each of four drives has an eccentric of least contact stress at 0.67 to 0.80 of its threshold; cut at 0.97, the peak pressure is 9% to 80% higher for a lever 21% to 46% longer.
A relief is sized for one torque
A cycloidal disc cut near its trimming threshold has long levers and a point on each flank, where the pins carrying most of the load press hardest. Relieving the edge a little near the flanks can take most of that point away. On a twelve-pin drive at 0.97 of its threshold, a relief 0.01 deep spread across a radian either side of each flank lowers the peak contact pressure by a quarter and keeps every lever, to within 7% of the disc cut at its least-stress eccentric, whose levers are a fifth shorter. The relief does it by unloading the flank, not by reshaping it, so its depth must be the contact's own approach at the design torque. At a quarter of that torque it is worse than no relief at all, and a relief of hundredths of the pin's radius, which is what was expected to work, takes pins out of the load.
Members that pull
Every link on this site so far has been rigid: it holds two points at a distance, in both directions, and a configuration is the root of an equation. A belt, a chain, a rope and a tendon are none of that. A strand has no shape of its own — where it runs is decided by the bodies it touches — and it constrains one way only, so it does nothing at all until it is taut, and a mechanism made with one has a different mobility in different places.
A member with no length of its own
Every link of a pin-jointed machine holds two points at a distance, in both directions, and a configuration is the root of an equation. A belt, a rope and a tendon do neither: a strand's path is decided by the bodies it touches, and its constraint is an inequality that does nothing at all until it is taut.
Where a strand leaves a body
A taut strand meets the surface it lies on at a right angle, and every book draws it that way. It is not a rule about strands: it is what being shortest looks like, and a family of paths that were never told about tangency has its minimum exactly there — 200.64346 mm against the construction's 200.64346.
The wraps add up to a turn
Every wrap angle in a closed run is computed on its own, from a pair of tangent lines that knows nothing about the others. Signed by which way the strand goes round, they add to exactly one turn — or to exactly nothing, for a crossed belt — and the integer is decided by the route rather than by any of the geometry.
A ratio that is a derivative of a length
A tackle is sold by counting the parts of line holding the moving block: four parts, four to one. Differentiate the strand's own length instead and a four-part tackle gives 3.927 with its blocks 220 mm apart and 3.617 at 90 mm — and the integer it is named for is a limit it reaches nowhere.
The strand that is slack
A rigid link removes a freedom wherever the mechanism stands. A strand removes one only where it is taut — so a point held by three of them has two freedoms in the middle of its region, one on an arc, none at a corner, and no single mobility count describes it at all.
The tensioner is the unknown
A linkage closes when a vector comes back to where it started: two equations, two unknowns. A strand closes when a number does — its length — however many bodies it runs over. So a run with one free body is determined, a run with two is not, and the arm angle that takes up 1,020 mm of belt is the root of one scalar equation solved to 1·10⁻¹³ mm.
The drum that is not round
The only thing about a body a strand can feel is the perpendicular distance from the axis to the tangent it leaves along. Ask for a rate and you have asked for that distance at every angle — and the shape comes back from it with no solve at all, unless the demand exceeds 1/(n²−1), at which point there is no shape.
The radius a winch works at
Line speed per turn is 2πr, and r is not a property of the winch. It is the radius of whichever layer is being wound, so a six-layer drum runs from 207.35 mm per turn to 395.84 — a factor of 1.909 with nothing about the machine changed, and a length that is quadratic in the turns rather than proportional to them.
A chain is not a strand
A chain has a smallest piece, and two things follow that no continuous model can have. Its pins sit on a polygon, so the radius that matters swings by 4.05% within every tooth of an eleven-tooth sprocket — and its loop must contain a whole number of pitches, so the centre distance that closes it comes in steps of 6.4834 mm.
One strand over many joints
Route a tendon over an idler centred on a joint's axis and the strand's length becomes an exactly linear function of the joint angle — 8·10⁻¹⁴ mm of departure over 203 degrees of travel. Move that idler 6 mm off the axis and the same drive's arm swings from 8.00 to 16.52 mm per radian.
The taut path has more than one answer
A strand from one point to another past a peg has two taut paths — 200.643 mm on one side and 219.165 on the other, against 200 mm of open air it cannot use. Both are shortest. Neither can become the other without passing through the peg, and no computation recovers which one was threaded.
Where a strand stops touching
A body joining a run costs length as the square of how far it intrudes — exponent 2.0000, measured over four decades — so at the moment contact begins the strand's length is stationary. That is why a tensioner set at the edge of its own contact takes up 0.000245 mm of belt per millimetre it travels.
A drum is a size, a wrap is a shape
A strand's whole behaviour is decided by where it leaves each body and how far round it goes, and both are angles. So a belt drive's velocity ratio, its wrap angles and its tackle's mechanical advantage transfer between drives of any size — and the one thing that does not is how much strand there is.
The wrap that walks along the axis
Every figure in this field is drawn in a plane, and a strand that goes round twice cannot be: the second turn has to lie beside the first. The plan view of the helix that results is a planar wrap exactly — so every wrap angle survives and the length does not, by five hundred parts per million on a real rope.
A strand in a tube
A Bowden cable's inner runs inside a sheath with a little clearance, and pulled it takes the shortest path the tube allows — a strand over pulleys of radius R − c at every bend. So its lost motion is the clearance times the total angle the sheath turns through: no bend radius in it, no route shape, and the S-bend that turns nowhere net loses as much as the U that turns back. Steering a handlebar changes it by exactly c times the change in angle.
Which walls a strand is held by
A Bowden inner is short of its sheath by the clearance times the total turning — a law with no bend radius in it and no route shape. It is exact while the inner touches the inside of every bend, and a bend shallower than the turn the inner spends crossing the bore is not touched at all. Past that the law is an over-estimate, and what the inner actually loses flattens onto a ceiling that has no clearance in it.
One belt for every step
A lathe changes speed by moving one flat belt between the steps of two stepped pulleys, so every pair of steps has to take the same belt. A crossed belt's length depends on the two radii only through their sum, so steps whose radii sum to the same value take the same crossed belt exactly — to the last bit of the arithmetic, over 266 random pairs. An open belt's length depends on the difference too: at constant sum it grows by d²/C, then d⁴/12C³, so an open belt made for the middle step of a five-speed pair is 12.3 mm short on the outer steps, and those steps must give up 3.74 mm of radius sum to take it.
A cone drive is not two cones
Take a stepped pulley to its limit and the belt runs on two drums, shifted continuously along them. Drawn, the drums are straight cones of opposite taper, and a crossed belt fits them everywhere because their radius sum is constant. But straight cones grade the speed unevenly — 0.58% per millimetre of shift in the middle, 0.72% at the ends — and an evenly graded drive needs drums whose radii are logistic curves. And an open belt, the common kind, fits neither: on a lathe-sized drive it is 12.3 mm short at the ends, and two identical drums that take it and keep the grading must each be shaved in proportion to their radius, 2.69 mm at the large end and 1.05 mm at the small.
A drum for every law
A cone drive's drums can be cut for any ratio law along the travel, not only the even proportion a machine tool wants. A crossed belt fits every law whose ratio stays positive, and one open belt fits any of them after a shave that depends only on the ratio at each station, so one table of shaves serves them all. What the law decides is everything else. Only the even proportion keeps the two drums one shape; the speed changing evenly or a fan's power changing evenly each need two different drums, 30 and 55 millimetres apart, and pack the change of speed into the slow end at three and fifty times the even law's rate.
A tolerance shortens the drum
A cone drive's drums must be long enough that a belt never straddles too large a change of ratio across its own width, so a demanded speed law sets the drums' length through its steepest point. Allowed to deliver speeds within a tolerance of the demand, a law's least steep version is a taut string through a tube: the demanded law with its steep end capped, exact everywhere else. Softening the speed law by 11% shortens its drums from 884 mm to 547, where the even proportion needs 300, and leaves it exact over three quarters of the travel. What softening never does, at any tolerance a designer would accept, is turn the law into the even proportion: that needs 52% for the speed law and 165% for a fan's.
A cam moves the law off the drums
A cone drive cut for a steep speed law needs long drums, because a belt cannot straddle a steep change of ratio across its own width. Cut the drums for the even proportion instead and let a cam between the handwheel and the belt fork place the belt, and every law runs on the same 300 mm. The steepness does not go away. The cam's groove is steeper than the even groove by exactly the factor the law's own drum was longer, 2.95 for the speed law and 44.7 for a fan's, and it is paid in handwheel turns instead of drum: six turns for the speed law against 884 mm of drum, and a fan law that needed thirteen metres becomes buildable.
Many of one thing
Every mechanism here so far is a chain: a handful of links, one or two loops, and a mobility somebody can check on the back of an envelope. A scissor lift, a folded sheet and a deployable ring are one small unit repeated, and three things change at once. The count of bodies is a parameter; mobility becomes the rank of a matrix rather than an arithmetic; and a unit that moves can be rigid the moment it is joined to another of itself.
Many loops, one freedom
A scissor lift, a folded sheet and a deployable ring are one small unit repeated thirty times, and three things change at once: the count of bodies becomes a parameter, mobility becomes the rank of a matrix, and a unit that moves can be rigid the moment it is joined to another of itself.
The loops are in the graph
Before a network is a mechanism it is a graph, and the one quantity that can be read straight off a drawing is how many independent loops it has: edges less nodes plus one. Grübler's count is that arithmetic and nothing else — which is why it is right about a tong at every size and says a deployable ring cannot open.
Each one moves, and together they do not
Take the pattern a Miura sheet folds along and move every interior vertex by a tenth of a panel. Every vertex still folds on its own — each is a spherical four-bar with a freedom of its own — and the four of them together fold to no angle at all, with a residual that starts at six millionths and never falls.
The freedom that survives repetition
A Miura sheet has one freedom at four panels and one at a hundred and forty-four, and the count runs the other way: plus one, then nought, then minus three, minus fifteen, minus ninety-nine. The gap between them is exactly (n − 2) squared, which is a hundred repeated constraints on a sheet with one degree of freedom.
Every vertex is a spherical linkage
Four creases through a point at fixed arcs from one another is a spherical four-bar — the object the spatial field is built on — with its link lengths printed on the paper as sector angles. The arcs hold to four parts in ten thousand million million at every fold, and on a flat-foldable vertex the half-angle tangents keep a ratio constant to nine figures.
A constraint that has been said already
Every constraint matrix leaves two null spaces, and a mechanism only lives in one of them. The other is the set of combinations of constraints that come to nothing — and its dimension is exactly the amount by which the count is wrong, on a deployable ring, a Miura sheet and a framework with twelve bars and six joints.
The ring that closes at every size
Two bent bars pinned at their kinks hold the angle between their connection lines at 135.000000° whatever you do to them, and two straight ones hold it at nothing. That is the whole difference between a scissor chain that grows in a line and a ring of eight that opens and shuts — and the count says the ring cannot move.
One input at one end
A lazy tong's reach is 2nL cos θ — exactly, to a part in a million million, over eight sizes and sixty openings — so it multiplies its input by the number of units. It multiplies everything else by the same number, including the part of the drawing nobody wanted multiplied: a unit cut a hundredth of a radian out puts a tong of thirty-two units 0.374 out at the far end.
It moves to first order and not at all
Two bars from one joint to two pinned ones, all three in line: the rank leaves a freedom pointing straight up, and lifting the joint stretches both bars. The obstruction is 1.414214, the walk travels a millionth of what it is asked to, and how far it gets is a property of the tolerance rather than of the mechanism.
Where the branches meet
A flat sheet is the one configuration every folding of a pattern passes through, and it is the one configuration where the rank is wrong about all of them. Three of a three-by-three Miura sheet's four apparent freedoms are not motions — and a grid that folds to no angle at all reports exactly the same four.
What a pattern has to satisfy
Move an interior vertex of a crease pattern and the folded state generally stops existing. How many conditions the drawing has to meet is not a matter of taste: it is exactly the number of dependencies among the constraints, measured at one, four and nine on three sizes of sheet, and a hundred on a sheet of a hundred and forty-four panels.
The cell that repeats for ever
Take the size of a network to infinity and it stops being a parameter. What is left is one cell, six bars, and a question nobody has to ask about a finite assembly: does the pattern's period count as a body? A square grid is rigid if it does not and shears if it does, and so does the kagome.
A stack that has to fit
A scissor stack's height is n·L·sin φ and goes to nothing as the bars lie down — on paper. The bars are made of something, and what stops the fold is two bosses meeting: every stage keeps 0.0949 whatever the stack does, which is exactly twice the boss radius.
Which diagonal rigidifies a grid
A three-by-three grid of squares needs five diagonals and eighty-one of the hundred and twenty-six ways of placing five will do. Which ones is not a rank question at all: it is whether a graph on the grid's columns and rows is connected, and eighty-one is the number of that graph's spanning trees.
The count says how many and not where
A kagome lattice has three joints and six bars in every cell and counts to exactly nothing, so a patch cut from it has as many mechanisms as its edge has lost bars: 5L − 5 for a rhombus of L cells a side, which the rank confirms at every size with no bar redundant. Straight or twisted, the number is the same. Where the mechanisms are is not: a straight patch keeps nearly half its edge weight in the middle, and a patch whose triangles are turned by 17° keeps a twentieth.
An edge that knows which side it is on
Cut a strip from a kagome lattice and each long edge loses the same bars, so the count says the two edges share the strip's mechanisms equally. Move each triangle's three joints by the same signed amount and they do. Reverse one move and every mechanism that is not a translation moves to one edge. The shift is a whole number per cell, set by the winding of one six-by-six determinant round the lattice's zone. It changes only where one family of bars lines up straight.
A seam with no bar missing
Join two distorted kagome lattices of opposite polarization along a line. Every joint on the line keeps all four of its bars, so Maxwell's count says the line holds nothing. It holds two per wavevector: two mechanisms one way up, two states of self-stress the other way up, exactly the difference of the two polarizations. The self-stresses are tensions the lattice carries with no load on it, along a line where no bar is spare. In a strip sixty cells wide the prediction holds at every wavevector a quarter of the zone or more from its centre.
A cable is half a brace
A square grid braced with rigid diagonals is rigid when a graph on its columns and rows is connected, so a three-by-three grid needs five braces and eighty-one sets of five will do. Brace it with cables instead, which pull and do not push, and each cable becomes one direction of an edge. The grid is rigid when that directed graph can get from every column and row to every other, which takes six cables, and only twelve of the 5,376 ways of placing six do it.
Twelve cables survive a slack one
A three-by-three grid braced with cables is rigid on six and never survives the loss of one while each cell holds a single cable. Let a cell hold two crossed and every one of the 262,144 fillings can be tried: none of eleven cables survives a slack one, thirty-six of twelve do, and the rule for which is a count at each row and column, two cables in and two out. That count is the whole rule on every grid narrower than four cells. At four by four it stops being enough, and a grid can meet it everywhere and still hang on a single cable.
A spare is worth what its pretension leaves
Thirty-six ways of placing twelve cables on a three-by-three grid survive a slack one, and the count at each node cannot tell them apart. Their loads can. Pretensioned at its best, every one of them takes a load of about 1.1 times its largest pretension before a cable goes slack. Lose a cable and the pull it carried is shared out elastically, and what survives is between a quarter and a four-hundredth of that. A pretension chosen for the loss instead of for the intact grid lifts the worst case: to 0.32–0.40 on the Latin squares, 0.22–0.27 when four cells are crossed, and never above 0.096 when six are. The family to build is the Latin square, and even among Latin squares three kinds of spare are not equal.
A grid that loses two crosses every cell
To stay rigid when any two of its cables go slack, a cable-braced grid needs three arrows into and out of every row and column of cells. On three by three that is eighteen cables, both diagonals of every cell, and no placement of seventeen survives. On four by four it leaves 576 placements of 24, and for once the count is the whole rule. Pretensioned for the intact grid, 570 of them lose a cable outright to some pair of losses. Pretensioned for the pairs, all of them keep something, and the bracings with no empty cell keep the most, up to 0.198 of the largest pretension.
Contacts that only push
Every constraint here so far has been an equation. A part resting against another part is not one: the contact says do not come closer and nothing at all about going away, so what the part may do is a cone rather than a subspace and a freedom stops being two-sided. Six constraints fix a body in space and six contacts fix nothing — the number is seven — and whether a part is held stops being a rank and becomes a question about where the origin sits inside a hull.
A constraint that only pushes
Every constraint in these essays so far has been an equation: a pin holds two points together, a bar holds two apart, a mesh holds a ratio. A part resting against another part says only *do not come closer* — so what it may do is a cone rather than a subspace, and whether it can move at all stops being a rank.
What one contact forbids
A rotation about a point is a twist, and a twist is affine in the point — so what a single contact permits is a half-plane of centres, with the boundary being the contact surface's own line. Reuleaux drew it in 1875 and it is exact rather than sampled, which is why every figure in this field is a picture of the plane rather than of a cone.
Four in the plane and seven in space
Six independent constraints fix a body in space and six contacts fix nothing, because d vectors can span d dimensions and can never positively span them. The minimum is one more than the dimension — and it is a floor rather than an answer: six contacts on a box held it in none of four thousand random arrangements and seven held it in twenty-one.
The test is a program, not a rank
Three independent routes to one yes-or-no: enumerate the escape cone's extreme rays by cross products, take the convex hull of the contact rows and ask where the origin is, or hand the whole thing to a simplex. They agree on every arrangement — and the first version of the third one reported a disc as held, which is the one part in the field that no number of contacts holds.
The escape is a place
A part that is not held escapes, and the useful thing is not that it escapes but where. The extreme rays of the cone are the corners of a region of the plane and its unbounded directions are translations — so the answer to 'this does not hold' is a picture with a shape, and the shape says where the next contact has to go.
Free to turn and unable to
An ellipse in a pocket the size of its own bounding box has four contacts whose rows span two dimensions, so the cone of permitted twists is a whole line and the first-order answer is that it spins both ways. It cannot turn by any amount whatever: the penetration grows as the square of the angle, with a fitted exponent of 1.9944, and a circle in the same pocket turns for ever.
The contact that is free not to touch
A hexagon on five contacts holds, and taking one of the five away leaves the margin at 0.0914 — unchanged, to every figure. That contact constrains nothing the others were not already constraining, and what it actually does is become the one member of the set that is free not to touch, with the decision made by errors nobody controls.
Free at every instant and going nowhere
Three points on a circle of 1.1 radii around a unit disc leave it free in every direction at every configuration — rank two, margin nought, the whole plane of centres shaded — and it cannot get out. The threshold is 1/sin(π/n), which is 1.154701 for three, and a flood fill of the free space agrees with the formula at every radius sampled.
Which way it comes out
Drop the rotation from the inequalities and the cone lives in two dimensions rather than three, so it can be drawn as an angle: a block in a vee has ninety degrees of directions out, a key in a slot has exactly one and no arc around it, and a dovetail has none at all. Three answers, and each of them is a different kind of joint.
Neither part comes out first
Two congruent Z-shaped pieces in a tray open at the top. Each has four contacts with the other, with normals at all four points of the compass, so each alone is blocked in every direction there is — and the pair lifts straight out. The removal cone of a set of parts is not built from the removal cones of its members, and *which part comes out first* is a question with no answer.
Where the jaws put it
Three jaws closing on a bar put its axis at −⅔ Σ h(u_k) u_k, which vanishes exactly when the section's support function is unchanged by a 120° turn. So a three-jaw chuck centres round, triangular and hexagonal stock perfectly and a square bar by up to 17.3 per cent of its own circumradius — and the workshop rule about symmetry that predicts this is wrong, because a six-jaw chuck centres a square.
Held is not located
Back every obstacle off by a clearance and the permitted poses become a polyhedron — bounded exactly when the arrangement is a hold, since an unbounded direction of it would be a ray of the escape cone. So whether a part is held is whether its pose set is finite, the clearance is what gives that set a size, and the two questions have to be settled in that order because no tolerance settles the first.
A cone has no size
What a set of contacts permits is a cone of twists, and a cone is closed under positive scaling by definition — so nothing about it changes when the part it holds is made bigger. Except that a twist is a screw, a screw has a pitch, and a pitch is a length.
Six hold nothing
Every exact-constraint coupling in these essays — Kelvin, Maxwell, three-two-one, and a Kelvin clamp with a seventh pad added — has rank six and holds the part not at all. The escape a Maxwell coupling leaves is a pure vertical translation with nothing else in it, which is not a defect: it is what a coupling is, and gravity is the seventh contact nobody draws.
Which contact to make accurately
A hold turns a set of contact tolerances into one linear inequality, and the weights in it are the coefficients of the combination that cancels — a quarter each on a square held by four, and 0.144 to 0.424 on a hexagon held by five. Above that line the part goes in and below it there is no pose it can take at all: not badly located, not out of position, no fit.
The hold is in the corners
A disc cannot be held by frictionless contacts and a regular polygon can, so a polygon with more and more sides has to lose its hold somewhere. Searched exhaustively, the best four contacts sit at alternate ends of four edges a quarter-turn apart, and their margin is the half-edge sin(π/n) less a correction that falls as 1/n² — 74% of it at eight sides, 99.4% at sixty-four. The hold is lost as the side shrinks, not as its square, and it is carried entirely by how far a contact sits from its edge's middle.
Two jaws and a coefficient
Every hold in this field has been frictionless, and two of its essays say what that leaves out. Put friction back and a part is held by two contacts in the plane instead of four and by three in space instead of seven; a disc, which no frictionless fixture can hold at all, is held by two jaws at any friction above nought; and an odd polygon needs a friction of exactly tan(π/2n). What friction takes away is the thing that made a frictionless hold a fact about geometry: a frictional hold is a hold per unit of squeeze.
A hold the load squeezes shut
A cam cleat bites harder the harder the rope pulls, and a wedge dragged by a bar into a narrowing gap squeezes the bar harder the harder it is drawn. Neither needs a screw: the load supplies its own squeeze, if the geometry lets it. A cam on a free pivot can push only along the line to its pivot, so it holds exactly while that line leans from the contact's normal by no more than the friction angle. A wedge's squeeze passes through two contacts, and it jams while its angle is no more than the sum of their friction angles — twice the figure usually quoted. And the set of loads either one holds does not shrink towards its threshold. It is lost there, all at once.
A stiff pin buys grip with release
A cam cleat on a frictionless pin holds a rope while the line from the contact to the pivot leans no more than the contact's friction angle. Give the pin friction and the cam's force may miss the pivot by the pin's friction circle, so the cleat holds to a lean steeper by exactly the angle that circle subtends: tan γ ≤ μ + (ρ/h)√(1 + μ²). But the same friction makes a cleat leaning less than arctan(ρ/h) lock against a backward pull as well, and refuse to let go. Both edges of the working window rise, the lower at a slope of one and the upper at √(1 + μ²): the grip gained is almost all paid for in release.
A cleat set hard lets go easiest
A cam cleat on a frictionless pin is freed by a lever with no effort at all. With pin friction the lever must beat h(T − N tan γ) + ρ√(T² + N²), and since a stiff pin lets the squeeze N sit anywhere in a band, the effort is set by where in the band it sits: nought at the top, twice the pin's friction moment at the bottom. A rising load drags the cam in to the bottom, so a cleat holding the load it was set by is the hardest to free — 0.412 times the pull times the pivot's height at a lean of 10° — while one jammed hard by hand, or one whose load has eased, frees itself. Below the backward-lock lean nothing jumps in the effort. What changes is the state: the slack cleat keeps the squeeze its largest pull left.
A cleat on a swinging pin has two leans
Carry a cam cleat's pin on a sprung link and, while the pin holds, cam and link turn together about the link's pivot: a second cam, frictionless, whose squeeze per unit pull is the cot of its own lean softened by the spring against the rope. A rising load is met at the pin's edge and a falling one at the link's, so a slack cleat keeps only what the link falls short of the edge — 12.5 times the pull on a fixed pin at 2°, 6.3 on a link at half the edge, nothing on a link at the edge itself. Nothing in it depends on the load's size: every force doubles with the pull.
The chain before the lengths
Every field before this one is handed a mechanism and asked what it does. Here the connections are the unknown: which graphs of links and pins are mechanisms at all, how many there are, and which of them are the same mechanism drawn twice. The search space is finite and every quantity is a count — one four-link chain, two six-link, sixteen eight-link, two hundred and thirty at ten — and the counting rule everything else on the site rests on turns out to admit eight graphs for every one that deserves it.
The mechanism is the graph
Twenty-one fields of this collection have been handed a mechanism and asked what it does. Take the mechanism away and keep only which link is pinned to which, and there is still a finite list of answers: one chain of four links, two of six, sixteen of eight, two hundred and thirty of ten — and 1,878 graphs at ten links that pass every count and are not among them.
Same links, same pins, different machines
Watt's six-bar and Stephenson's have six links, seven pins, four binary links and two ternary ones. Every count anybody can make on them agrees. They are different chains, they give two mechanisms and three, and the difference is whether the two ternary links share a pin.
What a count cannot see
At ten links, 1,878 graphs satisfy Grübler's rule and 230 are mechanisms. The other 1,648 contain a subchain that is already a structure — and on 1,165 of them the count says one degree of freedom, the rank of the constraint Jacobian says one degree of freedom, and both are right about a mechanism that does not have ten links.
Right until the size nobody checked
The characteristic polynomial of a chain's adjacency matrix is a fingerprint that costs nothing and separates every six-link chain and every eight-link one. At ten links it fails on two pairs — and on one of them, counting the ternary links tells the two chains apart while the polynomial does not.
Deciding that two chains are one
Two chains are the same chain when a relabelling of the links carries one to the other. Ten links admit 3,628,800 relabellings, and the census asks the question 26,335 times — so the answer is not a search but a rule that picks one labelling out of the graph itself, and asking whether the two strings match.
Which link to bolt down
A chain is not a machine until one of its links is held still, and which one is a decision. Two links give the same machine exactly when a relabelling of the whole chain carries one to the other — so the number of mechanisms a chain gives is a count of orbits, and the classical five six-bars and seventy-one eight-bars are that count.
Eight ways to drive it, and one machine
Bolting a link down is half the decision; the other half is which link carries the input. A four-bar has eight frame-and-input pairs and exactly one of them is a distinct machine — and across the eight-link census 320 listed pairs collapse to 153.
What has to be solved together
Hold a link, turn a neighbour, and the rest of a mechanism comes apart into the smallest sets that can be positioned one after another. Every set of two links is two circles meeting — a quadratic, two branches, no solver. A set of four is a system, and the Newton solve stops being a convenience.
Four that a compass cannot reach
Twelve of the sixteen eight-link chains can be positioned two links at a time, from at least one choice of frame and input. Four cannot be positioned that way from any of their twenty choices — and at ten links ninety of the two hundred and thirty are in the same position.
The candidates a search throws away
The obvious enumeration generates every labelling of every chain and keeps one. At eight links that is 8,494 complete graphs for 71 answers; at ten it does not finish. One rule — reject the labelling that a swap of two equal links would improve — takes it to 3,000 candidates for 1,878 answers in half a second, and twelve links is still out of reach.
Eleven assortments and four that are empty
How many links carry two pins, how many carry three, how many carry four: two lines of arithmetic admit eleven answers at ten links. Seventy-eight graphs have degrees the last four of them describe, every one of those graphs satisfies Grübler's rule exactly, and not one of them is a mechanism.
A catalogue is a search space
Dimensional synthesis searches over lengths within a topology, and the topology is chosen first — usually from memory, usually from a list of five. With a census the list is two hundred and thirty, every requirement that reads only the graph is a filter on it, and the choice stops being a habit.
The chain has no lengths
Every number in this field survives multiplying every link by a different scale factor, because there are no lengths to scale. Which is also the statement of what a census cannot decide — and the sixteen eight-link chains, each given one arbitrary set of dimensions and driven, produce a chart in which nothing belongs to the chains.
A machine with one dyad in it
Two hundred and forty joints, and two hundred and thirty-eight of them can be placed one at a time from parts already positioned. The whole of what has to be solved simultaneously is a single pair — the arm and the parallelogram carrying its second angle home. Size and structural depth are different axes, and a compiled machine is extreme on one and trivial on the other.
A graph has no numbers at all
Every other field of this collection has parameters a measurement could try to recover. This one has none. A chain is a graph, a graph is a set of links and a set of joints, and there is nothing about it that a scaling touches, a tolerance perturbs or an instrument determines.
A slide turns nothing
Make one joint of a chain a slide instead of a pin and the graph has a second decision in it before any length exists. The symmetries that counted mechanisms count these too — Watt's chain with one slide is three chains and eleven machines — and two facts read off the graph say which placements still work: a loop of slides alone is freer than the count, and a pin in a group of links the slides hold at one orientation cannot turn. Across 102 placements on the three smallest chains, both agree with the rank of the constraint Jacobian.
A contact turns and slides at once
A cam on its follower, a gear tooth or a pin in a slot takes one freedom away where a pin or a slide takes two, and it turns as well as slides. Give every joint of the four-bar, Watt's and Stephenson's chains a third label and there are 1,317 different chains. The rule that read slides off the graph extends to all of them, agreeing with the constraint rank on every one, and the two obvious shortcuts miss 36 and 769. What the graph cannot record is which way a contact's normal points — and on these chains that never matters, while a single stop added to a four-bar holds everywhere except through one point.
A stop aimed at its pole holds by curvature
A stop whose normal passes through its pole forbids nothing at first order, and whether it holds at all is decided by curvature. For a round nose the deciding curve is not the one it touches: the seat that just holds is the path of the nose's centre moved out by the nose, 1.193 on a four-bar where the contact point's own circle says 0.776. Past it, one contact holds both ways — loosely, as the square root of its clearance.
A seat with friction stays where it is pushed
A stop aimed through its pole holds a four-bar by curvature, because its normal has no lever about the pole. Friction has one, and the obvious guess is that it lets such a stop hold at first order. It does something narrower and stranger. It cannot hold what curvature releases, since a gap that opens carries no force for friction to be a share of. It makes a seat that holds up to twenty-six times stiffer, turning its cube-root give into a square root. And it makes any stop whose pole lies inside its friction cone stay where a load pushed it: released, the seat at the pole does not come back.
Return and stiffness come from the same aim
A stop with friction looked like a trade: aim it near its pole and friction multiplies its stiffness seventeen times, aim it outside the friction cone and it springs back when released. The trade is an artefact of how the gain was counted. Friction's factor multiplies a stiffness that vanishes at the pole, and measured outright a stop aimed further away is stiffer at every friction — 271 per radian at 1.1° and 104,000 at 21.3° with none, 4,631 and 184,000 at a coefficient of 0.3. Return and stiffness are bought by the same aim. What the aim costs is the other side: a face aimed off its pole holds nothing the other way, so a stop that must return both ways is a V of two faces, and across a friction band from 0.1 to 0.2 a V aimed 12.3° each way returns from both and is 8,800 times stiffer than the seat at the pole.
A V pressed onto its nose has a window
A V of two faces holds a four-bar's coupler both ways and returns it, as long as each face is aimed off the coupler's pole by more than the friction angle. Squeeze the nose between both faces and two different things can happen. A rigid V cut a hundred-thousandth tight cannot lock, because the crank moves the nose across the V and never into it; it only stops short of home, inside a dead band that is μ/tan ψ of the play an equally loose fit would leave. A V pressed on by a spring, as a detent is, rides back along its own axis, and that is the direction a wedge locks in. It returns from an aim of arctan μ at the contact and locks at 90° less arctan μ about its axis, so its aim has a window, and the V that is most nearly reversible inside it is a right angle.
The curve as an equation
Every other field here is handed its demand geometrically: three positions, a sampled path, a ratio. This one is handed a polynomial, and the mechanism is compiled from it rather than searched for — every monomial becomes a cosine, every cosine becomes a link, and the linkage closing is the equation being satisfied. The answer is exact everywhere the machine moves, and what it costs is not accuracy but size.
A demand that is an equation
Every field of this collection is handed its demand geometrically — three positions, a sampled path, a ratio at each angle — and hands back a mechanism that is right at those places and approximately right between them. This one is handed a polynomial, and the mechanism that comes back satisfies it everywhere it moves.
Every curve is a sum of cosines
Put the two angles of a two-link arm into any polynomial in x and y and what comes out is a constant plus a finite sum of cosines of whole-number combinations of them. Nine curves, three hundred random angle pairs each, and the two routes agree to 1.8 × 10⁻¹⁴.
A circle costs one term
A line expands to two cosines, a circle to one, a general conic to six. A lemniscate is degree four and costs five; a general cubic is degree three and costs eight. What a curve costs is not its degree — it is how many frequency pairs its own symmetry fails to cancel.
Four bars that add two angles
A rhombus on two links from one pivot points along the bisector of their angle, exactly, because a rhombus has equal sides. Hold its far vertex on a line and it reflects instead. From those two facts come negation, doubling and addition — and every whole-number combination of two angles a compiled machine needs.
A parallelogram carries an angle, and only so far
A direction computed at the frame is no use where it is needed. A parallelogram carries one from point to point — but only between two points a fixed distance apart, and that single proviso is what makes a compiled machine quadratic in the number of terms and turns most of it into transport.
Doubling is cheaper than adding
Multiplying an angle by eight costs three gadgets and multiplying it by seven costs six. The cost of an integer multiple follows the binary expansion of the integer and not its size — which is why the arithmetic in a compiled machine grows like d log d while everything else grows like the fourth power.
The machine, compiled
Twenty bars, twenty joints, and one degree of freedom. Every position is a converged solve on thirty-five equations, none of which mentions the polynomial — and the polynomial at the tracing point reads 1.3 × 10⁻¹⁴ across the whole working arc.
Five bars for a line, four hundred for a quintic
Nine curves compiled and counted: a line at five bars, a circle at eleven, a lemniscate at fifty, a general quintic at four hundred and thirteen. The growth is a fourth power of the degree, and three quarters of the largest machine is not computing anything at all.
The proof drew more than the curve
Sixteen ways to assemble one linkage. Eight of them close. Four put the tracing point on the curve and four put it somewhere else — at a closure residual of 9.6 × 10⁻¹⁵, which is the same floor the right ones reach. No tolerance on the closure could ever have told them apart.
A bar between two midpoints
In a parallelogram the midpoints of two opposite sides are exactly one side apart, and in the crossed assembly they are not. One bar between them admits the first and refuses the second — and it is one redundant equation per parallelogram, added on purpose, on a site whose constraint field is otherwise about overconstraint arriving by accident.
Where the machine stops being the function
Drive a reflector through the angle at which its rhombus flattens and it comes out computing something else. Nothing breaks: every bar is the length it was, the closure residual stays at 8 × 10⁻¹⁴, and the machine goes on turning. That is why every compiled machine in this field works over an arc and not a turn — the quintic's over a tenth of a radian.
Exact costs more than close
Watt's four bars are wrong by nine per cent of their stroke and Chebyshev's by twelve. Peaucellier's seven are exact to 4 × 10⁻¹⁶, and a compiled machine is exact to 4.8 × 10⁻¹⁴ in five. There is nothing in between — adding bars to an approximation does not walk down the axis, and the four-bar in every beam engine ever built is on the wrong end of it.
What universality is worth
The linkage exists, it is four hundred and thirteen bars, and it draws ten degrees of its curve. All three are true and only the first is in the theorem — which is the ordinary shape of a result about what exists, and the reason it was worth building one to find out.
A compiled machine and its own scale
A linkage compiled from a polynomial has bars whose lengths are the polynomial's coefficients and joints whose angles are its phases. Scale it and every coefficient scales — so the machine computes the same polynomial multiplied by a constant, which is a different polynomial with the same roots.
The price is on the equation
A line costs five bars. The same line, written as its own equation multiplied by a factor that is never zero, costs fifty — and the machine compiled from the longer equation draws the same line just as exactly. Every cost this field quotes belongs to a polynomial and not to a curve, and the cheapest equation of a given curve is a quantity nobody here has.
Two circles for the price of one
Search every multiple of a curve's equation by a polynomial of degree two and the cheapest is the curve's own equation, on four curves and by exhaustion. On the circle a second multiplier ties — and what it describes is two concentric circles, whose squared radii sum to four times the arm's link length squared, at exactly the cost of one.
The cheapest curves at each degree
Search every equation of degree up to six, not the multiples of one curve, for the fewest cosines a compiled machine has to build, and the answer does not depend on the degree at all: one term at every even degree, two at every odd one, each reached by exactly one set of frequencies. The cheap curves are concentric circles placed where Chebyshev's polynomials put them — k of them for the price of one — and, at odd degree, a line through the pivot with circles that come free. One symmetry of the arm forbids anything cheaper: its two links can be swapped without moving the tip.
Four terms of eighteen draw the quintic
A quintic whose equation costs a compiled machine eighteen terms is drawn to 1.4% of the arm's reach by four of them and to 0.02% by ten, found by trying every set. The four are not its four largest. They are a line through the pivot bent by the square of the radius, the family the cheapest curves come from, and a curve close to that family is nearly free. An ellipse, a parabola, a lemniscate and a folium are not close to it, and each costs its whole equation to come within four per cent.
Moving the pivot only adds terms
A compiled machine's cost is the number of cosines its curve's equation expands into, and moving the arm's pivot gives the same curve a different equation. So an expensive curve might only be expensive because of where the arm was put. Searched over 1,681 pivots, it never is. Moving the pivot adds the equation's own derivatives at the pivot as new terms, and they cancel only on one locus per curve — a centre, or a line — on which the catalogue's pivot already sits. Anywhere else the ellipse costs 6 instead of 4, the folium 12 instead of 11, the lemniscate 9 instead of 5, the parabola its 6 everywhere, and the quintic that four terms nearly draw costs 30 instead of 18.
The quintic's cheapness runs along a line
A quintic whose equation costs a compiled machine eighteen cosines is drawn to a few per cent of the arm's reach by four of them, and its exact count jumps to 22 or 30 the moment the arm's pivot moves. Priced by the nearest curve instead, the two directions part completely. Along the line x = 0 the nearest four-term curve stays within 7.3% of the reach out to 0.4 of a link and the nearest six-term within 3.4%. Across it the four-term price goes from 3.2% to 18% between a twentieth and a tenth of a link, and from 0.15 on four terms buy nothing over three. The move along adds four cosines of amplitude 0.2 and 0.4 times the move; the move across adds twelve, sixty to ninety times larger in total.
The quintic's cheap pivots lie off the line
Four cosines draw the quintic within a few per cent of the arm's reach from its catalogue pivot, and the earlier measurement found that cheapness surviving along the line x = 0 and dying across it. Mapped over a grid of pivots, it does neither exactly. The cheap pivots form a narrow valley tilted ten degrees off that line, about a twentieth of a link wide. In the valley four cosines hold the arc to 1.4–2.5% at every height out to 0.4 of a link; on the line itself, at the same heights, they miss by 6–7%. At two valley pivots every set of four cosines and every set of six was tried, and none beat the bent line. The amplitudes of the cosines a move adds predict a tilt a seventh as large, so they do not explain it.
Links with a width
Every link in every other field is a distance between two points, and a distance cannot collide with anything because it is not anywhere. Give each one a body and the constraint stops being an equation in the loop and becomes an inequality between pairs of parts — not local, not smooth, and not in the plane the mechanism is drawn in. Not one machine here fits in a single plane, a clearance turns out to be a function with corners, a sampled sweep needs a bound to prove it missed nothing, and a four-bar turns all the way round exactly when it cannot be built with a bearing at each pivot.
A link that takes up room
For twenty-three fields a link in these essays has been a distance between two points, and a distance cannot collide with anything, because it is not anywhere. Give every link a body and a question arrives that none of the constraint equations can ask.
A gap is a number
A collision test that answers yes or no cannot say by how much, and therefore cannot say what would fix it. The quantity this field is built on is one signed number: positive is a gap, negative is how far the parts would have to be moved to stop overlapping.
A shape with a dent in it
The separating-axis theorem is not approximately right about a non-convex shape; it is wrong, and it returns a confident number while being wrong. The repair is to cut the shape into convex pieces — and the shortcut everybody takes instead adds a hundred per cent more material.
A gap with corners in it
The clearance between two parts is a function of the crank angle, and it is not a smooth one. It has a corner wherever the closest pair of features changes hands, so its minimum is not where a derivative vanishes and is not at either end of the travel.
A sweep that missed nothing
A swept clearance check looks at finitely many positions of a machine that has infinitely many, and cannot report what it did not look at. Here is a twelve-sample sweep declaring a machine clear by 0.007 while it is 0.010 inside a stud — and the bound that refuses to certify it.
Two bars that have to cross
The site's own four-bar has its coupler inside its frame by a full link width for the whole of a turn. It is not an impossible mechanism; it is a mechanism that cannot be built in one plane — and the plane it has been drawn in for twenty-three fields was a convenience nobody had to pay for.
A plane is a colour
Assigning links to parallel planes so that no two conflicting parts share one is a graph colouring, and the answer for a four-bar is three. Then the pins have to get through, and the problem stops being a colouring: of the six proper three-plane colourings, two can be built.
The room a machine sweeps
Every point any part of a machine occupies at some position of its drive. It is a region rather than a curve, its area is an integral computed two ways, and the one shape in the field with a closed form is what the grid is calibrated against.
The hole the machine needs
A four-bar's joints fit in a box five units by four. Its material needs a box twenty per cent larger in area, and fills barely half of it. Both numbers are design quantities, and until a link had a width neither could be stated.
The crank that cannot turn all the way
Grashof's inequality says which four-bars turn fully. Ask instead how wide their links may be with a bearing at each ground pivot, and the same inequality answers the opposite question: every four-bar that turns all the way round sweeps a link straight over one of its own pivots.
Free space comes in pieces
Every arc in these essays has ended at a configuration the mechanism cannot reach. Put two studs in a four-bar's way and its drive falls into two arcs whose ends are configurations it reaches perfectly well and cannot occupy — and no quantity the solver computes tells one arc from the other.
A pin is not a point
A joint in the fields before this one is a name and two coordinates. A pin is a cylinder with material round it, a length through the stack of plates, and a head — and every one of those turns some construction that returns points into a construction that may return nothing buildable.
A body is all size
Twenty-five of the fields before this one compute quantities that are mostly shapes, recoverable from an angle sensor and transferable between machines of any size. This one computes clearances, footprints and swept areas, and not one of them is a shape — which makes it the only field whose whole output needs a ruler.
The gap is a straight line in the metal
Thickening every link by the same amount subtracts the same amount from every clearance, exactly, and moves the angle at which the worst one occurs by nothing at all. So a whole swept check can be done once on bars of any width and every other width read off by subtraction — until the closest pair changes hands, and never past zero.
The regions overlap and the parts never meet
A swept region is a projection along time, and a projection cannot be undone. Chebyshev's two arms share thirty-six per cent of the ground the smaller of them covers and never come within twice a link's width of each other — a false alarm the region test cannot avoid, and one it cannot make at all against anything that stands still.
A link may be bent
A link is two pins at a fixed distance and the metal between them is a free choice. Bending it moves no joint of the mechanism by more than 10⁻¹³ and moves the clearance by a tenth of a link length — enough to build a machine that a straight bar refuses, and worth exactly nothing against a bearing pedestal the link sweeps over.
Four arcs and the false alarm is gone
A swept region forgets when, so two parts whose regions overlap may never meet: Chebyshev's two arms share 4.66 square units of ground and never come within 0.38. Cut the drive into arcs, sweep each part over each arc, and compare regions only arc against the same arc. The test stays sound and costs the same sweep, and it forgets less with every cut. Four arcs clear Chebyshev's arms and eight clear Peaucellier's worst pair. A certificate that needs no grid asks for three times as many, and the same arms made wide enough to touch are refused at every number of arcs.
Three arcs, cut in the right places
Equal arcs clear Chebyshev's two arms at four and Peaucellier's long arm and rhombus bar at eight. Each cell of ground the two parts share is visited by each part in one unbroken stretch of the drive, so each asks for exactly one cut, somewhere between the two visits — and the fewest cuts that answer every cell are found greedily from one sweep and proved least by the intervals that forced them. Three arcs clear Chebyshev's arms and five clear Peaucellier's pair, and halving only the arcs that still flag finds four and five at three and four times the sweeping.
Four cuts serve the whole cell
A region test between moving parts needs its drive cut where each pair's two visits to shared ground separate, and one pair of Peaucellier's arms needs four cuts. The cell has six such pairs, and cut one at a time they would need sixteen. Cut together they need four, the same four moments, because every pair crosses its shared ground on either side of them. Built from a whole-drive region test the cell would take six planes; the four cuts give back the true four, where equal arcs need eight to do it.
A cam machine is cut at its handovers
In one symmetric linkage, every pair of parts crosses its shared ground on either side of the same few moments, so the whole machine needs no more cuts of its drive than its worst pair. Four tools driven by four cams into one work station do not share their moments. Each pair needs one cut, the pairs need six between them, and the machine needs three: one for each time the station changes hands. Each cut can move only within the time the station stands empty, which is the cam diagram's own timing margin, and when a schedule is squeezed until that margin closes, the tools collide.
Two stations share a cut when they hand over together
A cam machine's region test needs one cut of its drive for every time its shared ground changes hands. A cell with two work stations has two pieces of shared ground, each with its own handovers, and a single cut can serve both if it falls when both stations stand empty. So the cell needs somewhere between its busier station's count and the two stations' sum, and its cams decide where. Out of step, a press-and-transfer cell needs four cuts; with the same tools and strokes put in step, two. The transfer arm that carries the part between stations makes one shared cut for free, because while it swings it is at neither.
What a joint is
Every field before this one declares its joints and then counts what they take away. A count cannot tell a pin from a slide: both are one, and the two mechanisms you get by swapping them are not related at all. What a joint permits is a set of displacements closed under composition — a group — and the six lower pairs turn out to be the six groups a surface can have as its own symmetry. The same instrument, run on a whole mechanism, separates the two kinds of overconstraint by an integer.
A joint is a surface that slides on itself
Twenty-two fields of this collection have declared their joints and then counted what those joints take away. A count cannot tell a pin from a slide — both are one. What a joint actually permits is a set of displacements closed under composition, and it can be computed from the shape of the surface: one linear condition per point, and the answer is a null space.
Six, and no others
Eleven surfaces were handed to the same computation and six groups came out. A cone, a torus and an ellipsoid of revolution give the same joint; a scalene ellipsoid gives none; and the list does not grow when more surfaces are added, because a surface's symmetry group has to leave a two-dimensional set alone and only six groups can.
The count cannot tell a pin from a slide
A revolute, a prismatic and a helical pair all take five freedoms away in space and leave one. Grübler adds the same number for each, the constraint rank measures the same number for each, and the three joints have nothing whatever in common — one sends a point round a circle, one along a line, and one along a helix at a rate the joint decides.
Twelve kinds of freedom
Every set of displacements that is closed under composition is one of twelve, up to where its axis points. Six of them are joints somebody sells. Four are motions a designer may perfectly well want and cannot buy at any price. And there is nothing at all of dimension five — checked here on twenty thousand random subspaces, every one of which generated the whole of the six.
Almost nothing is a group
Eighty thousand subspaces of the twists were drawn at random and closed under the Lie bracket. Above one dimension, not one of them was already closed, and every single one generated the whole of the rigid displacements at the first bracket. Two pins with parallel axes close at three; move one axis a hair and they close at six.
What a point sees
A group of displacements has no shape, so the only picture of one is what it does to something. Fix a point and the six lower pairs draw a line, a circle, a helix, a cylinder, a sphere and a plane — the six surfaces the pairs are made of. And two of the twelve sweep the same surface and are still different groups, which is the honest caption on the whole method.
A chain multiplies
An open chain's displacements are the product of its joints' groups, one factor per joint, in order. Sometimes the product is a group — three parallel pins and a slide along them give Schoenflies motion, which is a SCARA arm and is why it has four joints. Usually it is not, and then the chain's poses are a four-parameter set that needs six numbers to describe.
Four joints that give a group, and four that do not
Two chains of four joints. Same joint types, same count, same mobility, same Jacobian rank, same everything measured here across twenty-two fields. One of them reaches a four-dimensional set of displacements that closes under composition; the other reaches a four-dimensional set whose logarithms fill all six dimensions. The difference is six hundredths of a radian in where two axes point.
Legs intersect
A serial chain multiplies its joints' groups and the product is almost never a group. A parallel machine's platform gets the intersection of what its legs permit — and an intersection of groups is a group, always, with no coincidence required. That is the only construction in this field that produces closure for free, and it is why a platform can be designed for a motion type instead of discovered to have one.
Two planes meeting in a line
Sarrus's linkage draws an exact straight line out of six pin joints, and the spatial field proved it by solving the mechanism sixty times and measuring a departure of 9.8 × 10⁻¹⁶. Here the same fact comes out of two planes and a cross product, with no mechanism solved anywhere — and the two routes are not redundant, because only one of them can tell you the linkage as built delivers it.
Compose two positions and see where you land
Take two configurations a mechanism actually reaches, compose the displacements that got it there, and ask what kind of thing the result is. A planar four-bar lands inside planar motion, to 4 × 10⁻¹⁶. Sarrus lands on its own line. Bennett's linkage lands three tenths of a radian outside the four dimensions its own displacements occupy — a one-freedom motion that generates all six.
The instrument that is not a derivative
Sample a chain of four random pins over a millionth of a radian and its displacements occupy four dimensions, exactly as a SCARA arm's do. Sample the same chain over two radians and they occupy six. The step is at 10⁻⁶, and where it sits is a fact about arithmetic while the two plateaux are facts about the mechanism.
A higher pair has no group
A disc resting on a straight edge may slide along it and may turn about its own centre. Both keep the contact exactly. Do one and then the other and the contact lifts off the edge by |t sin φ| — up to 1.2 radii over an ordinary range — so the two freedoms are real and the pair of them is not closed. The count is still right and there is nothing for it to be the dimension of.
The block in the guide has a length
A prismatic pair is a point constrained to a line, and a point on a line of length G has a stroke of G. A block of length ℓ has a stroke of G − ℓ, because both its ends have to stay on the rails — so a guide is as long as the stroke plus the block, and a third of a short one is not stroke at all.
A coupling that only translates
A coupling between two parallel, offset shafts turns its output at exactly the input's speed when, and only when, the relative motion of its two hubs contains no rotation — when it lies in the translation group. Oldham's two slides give that group by construction and so do two equal parallel cranks; a four-bar that is not a parallelogram gives the whole planar group and its output wanders by more than a radian. And Oldham's right angle is not what makes the ratio one: it is what makes the slides slide least.
Constant velocity is a mirror
Between parallel shafts a coupling keeps its output in step exactly when its hubs only translate relative to each other, which is membership of a group. Between shafts that meet, no group will do: a single universal joint at 30° and a correctly phased double Cardan joint both close to the spherical group, and one wanders by 8.2° a turn while the other keeps step to 10⁻¹⁵. What separates them is a mirror that swaps the two hubs. It either fixes every configuration of the joint or fixes none, so one symmetric position certifies a whole turn — which is what the phasing mark on a propshaft checks.
Two errors, one ellipse
A double Cardan joint keeps step only while a mirror fixes its motion, and any real one is a little off. Split each error by what the mirror does to it and only the half it reverses costs anything: a phasing error spreads the lag at sin a · tan a per degree, a difference between the working angles at tan a, and raising both angles together costs nothing at all. The two costs leave their marks an eighth of a turn apart, so they add as the sides of a right triangle — a root sum of squares that no assumption of independence was needed for.
A Rzeppa cage adds its clearances
A Rzeppa joint keeps constant velocity because its cage holds every ball in the plane that bisects the two shafts. Give the cage's windows a clearance c and the output can lag or lead the input. Six balls might be expected to limit that lag through the worst of them, or as a root sum of squares. They do neither. Opposite balls are one constraint, the cage's two tilts spend two of the three, and what is left adds the three pairs' allowances like clearances in series: 0.350 c at a 20° bend, twice what the worst ball alone would allow. The lag falls to nought each time a ball crosses the plane of the bend, grows with the bend itself, and a four-ball joint cannot hold it at all.
A floating cage frees three balls and not six
A Rzeppa cage that may slide along its own normal as well as tilt has a third freedom, and counting constraints against freedoms says a six-ball joint should then lose its hold on the lag altogether. It loses nothing: a slide moves both balls of an opposite pair the same way, so it spends one ball's clearance for every one it frees, and the six-ball lag stays at 0.350 c. A three-ball joint, whose balls have no opposites, loses its bound outright, because three ball centres always lie in some plane. Offset grooves then add a second limit, set by the worst ball alone, which takes over from the cage only past an offset of about 0.35 times the tracks' play over the windows'.
A tight groove leans on one ball
A six-ball Rzeppa joint's free play is set by the tighter of its cage and its offset grooves, and the two limits look alike from outside. They are not alike once the play is taken up. With every contact made a spring of the same stiffness and the joint twisted past its free lag, a joint whose cage binds spreads the first load across four to six balls' worth and, a third of a window clearance on, gives every ball between 0.17 and 0.19 of it. A joint whose grooves bind puts the first load on one ball alone over most of a ball pitch, and two clearances on it still spreads it over fewer than six. At the same torque its hardest-pressed ball carries 1.5 to 3.0 times the caged joint's. Tightening the tracks to cut the play buys the play with uneven loading.
A tight cage spreads the load until the tracks join in
A Rzeppa joint whose offset grooves set its play loads one ball first, and at equal torque its hardest-pressed ball carries up to three times a joint whose cage sets the play. Tighten the cage's windows below 0.559 of the track play and the cage becomes the limit at every input angle: the first load spreads over 3.6 balls' worth or more everywhere, the play falls, and at low torque the worst ball carries no more than a caged joint's. The cage does not stay in charge. Once the twist reaches the gap between the two limits the tracks engage, unevenly, and the worst ball climbs back towards the untightened joint's. A tighter cage widens the gap and pushes that torque up, so the windows' clearance is a choice about the torque at which even loading should end.
Numbers that were measured
Every field before this one takes the numbers on the drawing as given: chosen by a designer, cut by a machinist, and thereafter known. They are not known. Run the same kinematics with the parameters as the unknowns and the motion as the data and a new question appears with an exact answer — which of them can be recovered at all. A four-bar read by a protractor is a three-parameter machine however long you measure it, a coupler curve is drawn by three different linkages, a length is not toleranced because it is two holes that are, and the standard description of a robot arm breaks where the arm does not.
A dimension is a measurement
Every calculation in these essays takes the numbers on the drawing as given: chosen by a designer, cut by a machinist, and thereafter known. They are not known. This field runs the same kinematics with the parameters as the unknowns and the motion as the data, and the first thing that appears is a question with an exact answer — which of them can be recovered at all.
The matrix a calibration inverts
One row for every number an instrument reads, one column for every parameter that might be wrong. Every entry is a derivative the tolerance field has been computing since its first essay — so this field's central object arrived already built, and what is new is which way it is read.
The direction no protractor can see
A four-bar's output angle depends only on the ratios of its lengths. That is a sentence anybody would agree to, and it has a consequence with a number attached: the vector of the four lengths is annihilated by every row of the machine's own identification Jacobian, to 7.6 × 10⁻¹⁵, at every pose, for ever.
The coordinates the site already had
Three of a four-bar's four parameters are recoverable, so there are three recoverable quantities. They are Freudenstein's K's, which these essays have used to design function generators for as long as they have synthesised anything — and the same 3 × 3 linear system, read backwards, identifies a machine from three measured angle pairs.
A ruler and a protractor
What a measurement recovers is decided by the units of its readings. An angle is dimensionless and cannot see a size; a position is not and can. And putting both instruments on one machine recovers no more parameters than the better of them alone — it recovers the same ones six times better conditioned.
How many poses are enough
Three readings determine a four-bar's shape and the thousandth adds almost nothing. The rank is reached at three because there are three parameters, and everything after that is conditioning — which is a different quantity, improves for a different reason, and stops improving much sooner than anybody expects.
Where a calibration should measure
Choose each next pose to make the worst-recovered parameter as observable as it can be, and something happens that nobody asked for: the first three land as far apart as they can get, and every one after that bisects a gap. Nothing told the routine to spread them. It maximises a singular value, and spreading is what that turns out to mean.
What another measurement is worth
The observability of a four-bar's worst-recovered parameter goes 0.118, 0.162, 0.188, 0.208 — and then keeps going up by less and less until adding a pose changes the fourth decimal place. The flattening is not diminishing returns on accuracy. It is a space of fixed dimension being filled.
Four indices, four answers
Five numbers are in use for scoring how well a set of poses determines a mechanism's parameters. They are five different questions about one list of singular values, they rank pose sets differently, and the literature quotes the choice between them as a matter of preference. It is a matter of what the report has to carry.
The pose the machine cannot reach
A measurement plan is drawn against the nominal machine and executed on the real one, and the real one does not go quite where the drawing says. A pose that falls outside the travel returns no reading at all — which is not an error, not a failure of the instrument, and not nothing: it is a measurement of the limit position.
Every length wrong, every reading right
A four-bar was built out of true and measured at thirty positions. A calibration started from the nominal dimensions reproduces every reading to 1.8 × 10⁻¹⁶ radians and returns four lengths, not one of which is the machine's. They are the machine's, multiplied by 0.99229 — every one of them, to fifteen figures.
The instrument's error, multiplied
Repeat a whole calibration on independently noised readings at four levels three decades apart and the error in the recovered shape is linear in the noise, with a fitted slope of 0.9994 and a constant of 1.90. That constant belongs to the mechanism and the poses, not to the instrument — and it is bounded by one over the smallest singular value.
Reading a residual
A residual that falls to the instrument's noise and stops means the model is right. A residual that stops above it means something is missing, and which something can be read off how the leftover is distributed over the poses — as a constant, as a pattern in the crank angle, or as one bad reading.
A parameter the model has not got
The machine's tracing point is 0.198 units from where the model says it is, and the model has only four lengths with which to say so. It absorbs the discrepancy: the error over the measured half-turn falls by a factor of thirty-three, the error over the other half falls by twenty-one, and the rocker comes back eight per cent short.
One set of lengths, two machines
Three measured input–output pairs return a four-bar's four lengths to fourteen figures. Assembled the way the data was taken, that linkage reproduces every reading to 2.5 × 10⁻¹⁴ radians. Assembled the other way — which the same four lengths permit — it misses them by 268°, and no equation in the identification knows the difference.
Three machines, one curve
Roberts's theorem says every four-bar coupler curve is drawn by exactly three different four-bars. Read as an identification problem that is a least-squares objective with three separate exact minima, whose cranks differ by sixty per cent — so an instrument that records only where the tracing point went has three answers, and no amount of data chooses between them.
Nine parameters, two of them invisible
A Watt six-bar has seven lengths, a fraction and a ground pivot's two coordinates. Read by a protractor on its output link, its identification Jacobian has rank seven — and the second missing direction is not a scaling of the machine at all. It is a scaling of the second loop alone, about the pivot the two loops share.
A machine that measures itself
Put an encoder at each end of a one-freedom loop and every pose gives one scalar equation. The equation is Freudenstein's, it is linear in three unknowns, forty poses make a three-column least squares at a condition number of 8.76 — and no instrument outside the machine is involved anywhere.
Where a length comes from
A coupler 3.5 units long is a part with two holes in it, and neither hole's position is the length. What reaches the length is only the part of each hole's error the two do not share — so a drawing that tolerances the length is describing a part nobody makes, and is out by a factor of √2 in one direction or by everything in the other.
Where the two analyses cross
Tolerance the lengths and you get one number whatever the shop does. Tolerance the holes and you get a curve, running from 1.414 times that number when nothing is shared to zero when everything is. They meet at a shared fraction of exactly one half, and the crossing does not depend on the machine, the tolerance or which length is being asked about.
The same part, dimensioned twice
Three holes, one process, two drawings. Dimensioned as a chain the errors accumulate and the span is ±0.0141; dimensioned from a baseline the span is ±0.0100 and the gap between the second and third holes is ±0.0141. Exactly √2 apart, in opposite places, and nothing about the part or the process decides which — the drawing does.
Which feature to hold tight
A budget divided between four lengths in inverse proportion to their sensitivities gives one answer. The same budget divided between the features those lengths are derived from gives the same answer exactly — unless the parts are made differently, in which case the tightest tolerance moves from the rocker to the crank and the frame's loosens by a factor of two.
The common normal, and where it is
The Denavit–Hartenberg convention reads all four of its numbers off one line: the common normal between two joint axes. Two parallel axes do not have one — every perpendicular meets both at right angles — and two nearly parallel axes have one that is somewhere else entirely.
A number that runs away
A hundredth of a degree of unintended twist on a nominally parallel pair of joint axes puts the Denavit–Hartenberg offset at −1,102 link lengths. The extraction from the geometry and the closed form agree to 5 × 10⁻¹⁶ over three decades, and the worst case over the tilt is exactly A/2α.
The chart breaks, the machine does not
The same two axes, over the same three and a half decades of twist. In one description a parameter runs from 0.003 to 1,102; in another the condition number is 7.5501 and does not move in the fifth figure. A quantity that diverges in one chart and is constant in another is a property of the chart.
Six per joint is two too many
A joint transform is six numbers, and a six-joint arm with a base and tool frame is forty-eight. A measurement can distinguish thirty. The difference is not a saving — it is an eighteen-dimensional set of exactly equivalent answers, and a fit returns whichever member of it the damping prefers.
Which numbers have a size
Take every length in a mechanism up and down together and fit the power each computed quantity follows. A transmission angle lands on zero, a coupler point's speed on one, a path curvature on minus one, an enclosed area on two — and a tolerance band held to a fixed ±0.01 lands on minus one, which nobody would guess.
A count is neither
Every quantity in the scaling survey lands on an integer power — zero for a shape, one for a length, two for an area. A mobility lands nowhere. It has no dimension at all, it does not move under any perturbation, and the probe that sorts the rest of the site's numbers returns nothing for it.
A calibration is a synthesis with more equations
The site's second field prescribes three input–output pairs and solves a 3 × 3 linear system for a linkage. This one measures thirty pairs and solves the same system in the least-squares sense. Same matrix, same coefficients, same closed form — and the only structural difference produces every question this field is about.
Two instruments disagree about the worst
A protractor recovers three of a four-bar's parameters at a condition number of 5.2. A coordinate machine recovers six at 162. Neither number says which parameter is worst recovered, and when both are asked, they name different ones — because a condition number is a summary of a list and the list is what a report needs.
What a model is allowed to change
Before a calibration runs, somebody decides which numbers it may move. Leave one out and the fit absorbs it into the others; put one in that the instrument cannot see and the fit returns whatever the damping preferred. Both decisions are made before any measurement, both are checkable in advance, and neither is usually checked.
What this field cannot measure
Every field of this collection has a boundary and this one has three: a direction the readings cannot span, an alternative no derivative detects, and a model nobody thought of. The first is computable exactly, the second needs a search, and the third is not detectable from data by any method at all.
The problem backwards
Given the lengths, find the motion is the reader's problem. Given the motion, find the lengths is the designer's, and it is why coupler-curve atlases were printed and sold. The constructions are exact; whether what they produce can be built is a separate question.
The problem the other way round
Every essay before this one starts from link lengths and finds the motion. That is the reader's problem, because lengths are what a drawing shows. It is not the designer's problem, which is the reverse — and the reverse is hard enough that for a century the practical method was to look the answer up in a book.
Three positions, and a circumcentre
The whole of three-position synthesis is one observation: a moving point occupies three places, three points that are not in a line lie on exactly one circle, and that circle's centre is where the fixed pivot has to be. No iteration, no tolerance, and every point of the coupler is a candidate.
What the fourth position costs
With three prescribed positions every point of the coupler will do, and a designer is spoilt for choice. Add a fourth and the whole plane collapses to a curve — only points on a particular cubic have four images that lie on a circle, and the cubic is Burmester's.
Three linkages, one curve
Every coupler curve is drawn by three different four-bars, not one. The other two can be constructed from the first with a single complex multiplication, they have different proportions and different ground pivots, and the roles of their bars are permuted — what is a coupler in one is a crank in another.
Five positions, and what is left
Three prescribed poses leave a whole plane of choices. Four leave a curve. Five leave four points, and finding them is the first thing in the synthesis essays that a compass cannot do — it needs two cubics intersected, which is algebra rather than construction. Four points give six four-bars, and two of them can be built.
Three problems called synthesis
Prescribing a path, prescribing a whole pose, and prescribing a relation between two angles are three different problems with three different counts, and the word synthesis covers all of them. The third has a property the others do not — eliminate the coupler angle and the design equation becomes linear, so a four-bar that computes a logarithm falls out of a 3 × 3 solve with no iteration at all.
Where the precision points go
A linkage that matches a function at three points is wrong between them, and where the three points are put decides how wrong. Chebyshev spacing cuts the worst error by a third against even spacing, for free — and the reason has nothing to do with mechanisms. It is a fact about a polynomial the linkage has never heard of.
The linkage that is only nearly right
Stop demanding that a linkage pass exactly through three points, and ask instead that it be close everywhere. Three linkages result, none of them passing exactly through anything, and each is the best by a different measure — the least-squares fit beats the interpolant on average error and loses to it on the worst case. An optimiser gives exactly what it was asked for and nothing else.
Where an optimiser starts
An approximate synthesis is a local search on an objective that is non-convex, disconnected and not everywhere defined, so the answer depends on where the search began. Nothing in the optimisation supplies that. What supplies it is the exact constructions of four earlier essays, and an atlas of coupler curves — which is why a method superseded by computers is still the thing that feeds them.
How many points may be prescribed
Five poses, five angle pairs, nine points — three numbers that get quoted as properties of a four-bar and are properties of what somebody decided to count as free. Derive them instead, and the fifth precision point turns a linear solve into a system with 128 paths, twelve finite solutions, four real ones, and exactly one linkage anybody could build.
Choosing the chain before the lengths
Every synthesis method in these essays starts by assuming a topology, and the assumption is usually a habit. What the graph fixes before any dimension is chosen is the number of free parameters — two per pin less four — and therefore how many positions can be prescribed at all.
Prescribing a curve rather than points
A four-bar can be made to pass through nine prescribed points and no more; past nine the problem is over-determined and the answer is an optimiser's. Prescribe the whole curve as an equation instead and there is no counting to do — but the mechanism that comes back has four hundred bars where the four-bar had four.
What a synthesis assumes it knows
Every construction in this field is handed a demand in absolute coordinates — three positions of a coupler plane, at stated places — and returns a linkage in the same coordinates. Scale the demand and the answer scales, which means the construction's whole content is about shape and its answer carries a size it was given.
A defect that is not kinematic
The synthesis field's survey ends in three verdicts and all three are about which solutions a sweep visits. Here is a fourth, found by asking how much room two pins need — and it is the first defect here that a simulation cannot find, because in the equations a pin is a point.
Where a pin becomes a slide
Three-position synthesis gives every point of the moving body a fixed pivot, except the points whose three images fall in a line. Those want a slide, and they are not scattered: they lie on one circle, the circle through the three image poles, which a single line of algebra predicts and a contour of a measured length draws to 10⁻¹⁵.
The slide a fourth pose leaves
Three prescribed poses leave a whole circle of body points that want a slide. A fourth leaves exactly one, found where two of those circles meet a second time. It had been predicted as one or three, because a real cubic meets the line at infinity in one real point or three. But both of Burmester's cubics pass through the two circular points, so only one real point is left. Four hundred random four-pose problems each have one slide and one slot pin, both found with compasses.
The slide five poses almost have
Five prescribed poses leave no slide: a body point would need five images in a line, three conditions on two coordinates. But each of the five four-pose subproblems has exactly one slider point, and each misses the pose it left out by a distance a pair of compasses can find. Those five misses price the best compromise without solving for it. The slide that misses every pose by the same amount misses by one over the sum of their reciprocals — the misses add like resistances in parallel — and giving up the best single pose instead is always between two and four times worse.
The load picks the wall
The best slide for five poses misses each by 0.0140, on alternate sides, so a block with that much play either side reaches all five — if it bears on the right wall at each. Nothing in a slot says which wall. The load does: the block presses whichever wall the load's moment about the crank pin sends it to, and changes walls only where that moment changes sign. Four changes are needed. The body's own weight gives two, in the wrong places, and play then makes the slide worse; a working load gives one, and needs nineteen times the play.
A spring balanced against the load
The best slide for five poses reaches all five with play in its slot only if its block bears on alternate walls, and neither the body's weight nor a working load puts it there. A spring can, and whether one does is a linear program. No point of the body can carry it: for every such point there are weights on the poses under which any spring's pull cancels. A point on an arm below the body can, and a spring from it pulling at most 11.7 times the load gives + − + − + exactly. The spring alone presses one wall throughout; it works by balancing the load, and only for loads within nine per cent of the one it was chosen for.
The arm belongs to the poses
A spring from the frame can put a rocking slide's block on the alternate walls its five poses need, but only from points on an arm below the body, never from the body itself. That looked like a fact about the working load the spring was balancing, and the body's own weight, which never changes, might have moved the reachable points onto the body. It moves them nowhere. Under no load, the weight, the working load and both, the program finds the same 155 points of the body's frame, the same lightest body point and the same stiffness, and only the spring's attachment point moves. The arm belongs to the five poses. And the earlier spring's band of nine per cent on the working load was its design margin: the band grows with the spring's pull, about one unit of load per hundred of pull.
A constant pull holds the band for less
A spring with free length was the obvious way to make the rocking slide's spring cheaper, and it does nothing: offered one, the lightest design takes none, at every arm point tried. The spring's law never mattered to where it may hang, because the walls depend only on which side of each crank pin the spring's line passes, and that cuts the attachment points to a polygon whatever the spring is. What the law decides is the price. Three poses bind at the lightest attachment point, and a spring whose tension follows its length must pay for the longest of them. A pull that is the same at every pose pays for none of them, and holds the band from nought to the design load for 44.6 units against 52.0. Designing for the band directly halves the hundred units the earlier essay read off by scaling.
The other ways in
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