Six things a body is not

Six things that get said about links with a width. None of them is careless, every one is exactly right about something, and every one is answered here with a number from the gap curve, the layer census or the width search.

Two of the six mistake a verdict for a measurement, two mistake a picture for a configuration, and two mistake one kind of question for another that happens to use the same words.

Nought: the list itself #

This is the eighteenth of these lists on the site and the form has a hazard, so it is worth restating.

Every claim below is one somebody could reasonably hold after reading this field’s opening rung, and none is a straw position. They are the readings a careful person arrives at from a correct account, which is what makes them worth answering.

They are also, all six, positions the work behind this field held at some point while it was being built. The gap really was a predicate until the width search needed something to bisect on. The sweep really was believed until the stud machine was built to break it. A list of things the work got wrong is a more honest object than a list of things a reader might, and this one is the former.

One: a collision test answers whether the parts collide #

It does, and that answer is worth almost nothing.

A predicate cannot be plotted against a crank angle, minimised over a sweep, bisected on a width, bounded between samples, or compared with a second route — two implementations agreeing on false have agreed about nothing. Every result in this field comes from the number rather than the verdict.

The width search bisects because the worst gap falls monotonically as the links get fatter, which makes the crossing unique. A predicate gives a plateau of false with an edge somewhere in it and no gradient to walk. The tolerance study reports an interval of gaps over a box of lengths, and its whole content is where the interval sits relative to zero. The free arcs are the sign of the number read as a set.

And the depth is directly actionable in a way a verdict is not: −0.432 says the coupler and frame are fully crossed and need a plane between them; −0.02 says thin something by a fiftieth of a link width. The same verdict covers both.

Two: a convex hull is a safe approximation #

It is safe and it is not an approximation of anything useful.

A hull contains the part, so it can only report interference that is not there, never the reverse — a design that passes on hulls certainly passes. The question nobody asks is what the guarantee costs.

On a right-angled bell crank with arms 1 and 0.7 and a half-width of 0.08, the hull is 105% more material than the crank. More than double, and all of the difference is in the notch between the arms — which is exactly where a designer puts something, because it is the space the shape was chosen to leave. A block in that notch clears the real crank by 0.020 and is 0.120 inside its hull.

So the hull says no to the one arrangement the part exists to allow. The complaint is not that it is wrong; it is that on the arrangements anybody actually tries, it is wrong by more than the quantity being measured. A test whose answer is dominated by its own conservatism has stopped depending on the design.

Three: sweep finely enough and the check is conclusive #

Finely enough is not a property a sweep can report about itself.

A four-bar with a stud a tenth of a unit across bolted to its frame, swept at eight samples, reports a smallest gap of +0.081. At twelve, +0.007. At sixteen and beyond, −0.010: the crank is inside the stud.

The twelve-sample answer is the dangerous one. Its curve is smooth, its minimum is interior, its witness pair is sensible, and its margin reads like a tight but successful design. There is no signal in it that says a feature was stepped over — the samples either side of the encounter are half a radian apart and the crank was elsewhere at both.

Doubling and getting a different answer is evidence the first was too coarse, and the same argument applies to the second. What terminates it is a Lipschitz bound: every corner of every body is an affine function of its link’s two joints, so its velocity is the same combination of joint velocities the mechanism already solves for, and the largest corner speed V bounds how fast the gap can fall. At V = 1.256 and Δθ = 0.571, the least the gap can be between two adjacent samples is −0.281, and the certificate refuses. On the machine that does clear, at sixty samples, the same bound is +0.038 and the sweep is conclusive.

Three and a half: more samples is the fix #

A corollary of the third worth separating, because it is what people do rather than what they say.

Faced with the stud machine, the natural response is to raise the sample count until the answer stops moving — and here that works: from sixteen on, every refinement returns −0.010. The trouble is that stops moving is not observable in advance and is not a guarantee afterwards. A feature narrower than the new spacing hides exactly as well as the old one did, and the only evidence that none exists is that none was found.

The bound turns the same effort into a statement. Run it backwards and it says how many samples would be needed: for the four-bar with a post, V·Θ/g = 1.256 × 6.283 ÷ 0.104 ≈ 77 samples, and the machine certifies at sixty because the estimate is conservative. So the count is not a matter of judgement at all — it is a computation whose input is the clearance being claimed.

And it says the useful thing plainly: the sample count needed goes as one over the clearance. A machine with a comfortable margin certifies at a handful of positions; a machine that nearly touches needs an unbounded number; one that touches exactly needs infinitely many, because there is nothing to certify.

Four: if two links cross on the drawing, they interfere #

Chebyshev’s linkage draws with its two long arms crossing and clears by +0.380 over the whole of its working arc. It is the one machine in this field’s catalogue that could be built in a single plane.

A drawing of a motion has thrown the time away. Two bars that cross on the page have crossed paths; two bars that interfere have been in the same place at the same configuration, and a picture of a whole sweep cannot tell them apart. The same reading applies to a coupler curve that crosses itself: the tracing point was at that place twice, at different crank angles, and nothing collided.

That is why every gap in this field is computed at a configuration and then minimised, rather than read off a picture — and why the swept region, which is exactly a picture with the time thrown away, can only ever give a conservative verdict. An obstacle outside the region cannot be hit; an obstacle inside it may never be hit either.

Five: interference is a force problem #

Every number in this field is a distance between two polygons at a configuration this site’s solver produced. There is no force anywhere in it.

The confusion is real and comes from the words. Contact, clearance and interference all carry a load in most engineering writing, and this site has a whole field — holding — in which a contact is a constraint that shapes the motion, with a cone of admissible velocities and a rank.

Here a contact is a failure of the design: the mechanism is meant to move freely and the material is meant to keep out of the way. Two fields using the same word and meaning opposite things by it, distinguished by whether the touching is intended.

What is genuinely outside is what happens after two bodies meet. A negative clearance says the mechanism as dimensioned cannot be made of solid material; whether a real one would bind, spring, wear a groove or break is a force question and is where the boundary is.

Six: a planar mechanism is planar, and the planes are a detail of manufacture #

Not one of nine machines in this field’s catalogue fits in a single plane, and the four-bars need three rather than two.

The offsets really are not kinematic — every length, angle, position, velocity and rank is unchanged, and a planar mechanism assembled in parallel planes is the same planar mechanism it was. That half of the claim is exactly right. What is wrong is detail: which link goes in which plane is a decision with a right answer and several wrong ones, and it is forced by the material rather than chosen for convenience.

And the decision is not a graph colouring, which is the second half of the mistake. A pin joining planes 1 and 3 passes through plane 2, and a link in plane 2 whose material covers that pin is pierced by it. Of the six proper three-plane colourings of a crank rocker, two can be built. Of Peaucellier’s 192, exactly 96 can.

A designer who colours the conflict graph and takes the first answer has a two-in-three chance of a machine with a pin through a link — and no gate on this site would notice, because the colouring is proper and every link is in a plane of its own from everything it conflicts with.

Six and a half: the plane count is the depth #

The last claim has a quieter version that is also wrong, and it matters to anybody reading the tables.

A machine needing three planes is not three plates deep. It is at least three plates deep, plus running clearance between them, plus whatever retains the pins at the outside — a head, a circlip, a washer — none of which this field models. Every plane here is an index, not a height, and the count is a floor.

The one place a real depth is computed is the scissor stack, where the thickness is the whole question and is put in by hand: two planes at any stage count, and a stowed height per stage of exactly two boss radii, because what stops the fold is two bosses meeting rather than anything about the stack.

That is the honest shape of the field’s three-dimensional claims. The plane count is exact and is a count. The depth is a different quantity needing three parameters this field does not carry, and it is computed once, on the one machine where the answer is the point.

What the six have in common #

Four of them are the same mistake with different objects: reading an answer as though it were about a machine when it is about the drawing of one.

A verdict is about the test rather than the part. A hull is about a convenient shape rather than the part. A sweep is about the samples rather than the motion. A crossing on the page is about the picture rather than the configurations. In each case there is a correct statement underneath — the test really did return false, the hull really does contain the part — and the error is in what it is taken to be evidence for.

The other two are the field’s own boundary being read in the wrong direction: force where there is none, and detail where there is a decision. Both of those are the kind of thing a site draws a boundary essay to fix, and both are worth restating whenever a field arrives whose words are borrowed from an adjacent subject.

Where each of the six came from #

It is worth recording, briefly, which measurement caught each one, because five of the six were caught by a number contradicting another number rather than by anybody thinking.

The predicate was never believed for long: the width search needed something to bisect on within an hour of the field starting, and a predicate has no gradient.

The hull was caught by a figure. It was drawn to illustrate that a hull is conservative, and the number that came out — 105% — was so much larger than expected that the figure changed its argument.

The sweep was caught deliberately: the stud machine exists for no other reason, and it was built by scanning obstacle positions for one whose encounter was narrow enough to step over.

The crossing on the drawing was caught by Chebyshev’s linkage refusing to conform. Every other machine in the catalogue reads negative, and the one that reads +0.380 is the one whose arms visibly cross.

The force reading was caught by writing, not by measuring — it is the one item on the list that is a matter of what the words mean rather than of what the numbers say.

And the colouring was caught by a bug: the layer search reused the colouring search’s symmetry breaking, which is valid for a colouring and invalid once betweenness matters, and reported that a four-bar cannot be built out of solid material in any number of planes. A result that absurd is easy to catch, and it is what made the ordering condition visible as a separate thing.

The seventh, which is this list #

One more, and it is about the form.

None of these six is refuted by an argument. Each is refuted by a number that comes out of code in this repository: 105%, +0.007 against −0.010, +0.380, six against two. A list of misconceptions answered by reasoning would be a list of opinions, and this site’s standing objection to a great deal of published mechanism writing is exactly that it argues where it could measure.

There is one more property of the form worth defending. Each of these lists on the site is written at the end of a field, after the machinery is built and the results are in, and each of them is where the corrections end up — the places where the work was wrong and the measurement said so. That makes them the only essays here whose content could not have been planned in advance, and it is why the eighteenth one still has six entries rather than repeating the seventeenth’s.

The corresponding hazard is that a number can be wrong in a way an argument cannot, silently and confidently — which is why every one of the six above is produced by machinery with a second route attached to it, and why the certificate exists at all.

There is one more thing worth reading off the list, and it is about the direction each mistake errs in. A convex hull is conservative: it over-estimates the part, so a hull-based test reports interference that is not there and never misses interference that is. A sweep is optimistic: it samples, so it misses interference between samples and never invents any. Two approximations, two opposite biases — and a designer who uses both, as most collision work does, has an answer whose direction of error is unknown. Conservative on one axis and optimistic on the other is not a compromise; it is a verdict with no sign attached, and it cannot be argued with in either direction. The remedy is not to abandon either technique but to keep the bias attached to the answer: the hulls do not intersect at any sample is a statement with two qualifiers, and each of them points a different way. Reporting the pair rather than the verdict is what lets a reader know which way to be careful.