Field

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 crank rocker with a post: the closest pair at one position. The same four-bar with a post bolted to the frame, just clear of the coupler's path. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.2561 here, between rocker · post. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.

A link that takes up room

What changes when a link stops being a distance constraint and becomes material: an inequality between pairs of parts, not local, not smooth, and not in the plane the mechanism is drawn in.

Peaucellier's cell: the closest pair at one position. Eight links, ten pins and an exact straight line — the site's densest planar loop. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.1983 here, between long arm A · crank. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.

A gap is a number

The signed clearance between two parts, computed by closest features when they are apart and by the separating-axis theorem when they are not, with both routes calibrated against closed forms.

A dented shape, cut into 2 convex pieces. Ear clipping cuts the outline into triangles and Hertel–Mehlhorn then deletes every diagonal whose removal leaves both sides convex, which takes this crank to 2 pieces rather than the 4 the triangulation produced. The pieces tile the polygon to 1.8e-16 of its area, and — the part the areas cannot check — a point is inside the pieces exactly when it is inside the polygon, tested at 4,000 random points per shape with 0 disagreements. The gap between two parts is then the best answer over the pairs of pieces, which is why the decomposition has to be right rather than merely plausible.

A shape with a dent in it

Why a dented part cannot be tested by a separating axis, how ear clipping and Hertel–Mehlhorn cut one into convex pieces, and what the convex hull costs when it is used instead.

The gap is a function, and it has corners. The smallest gap over every tested pair, at each of 360 solved positions of the crank. Two things are visible that a check at the ends could not report. The minimum, 0.1041, is in the middle of the travel and not at either end. And the curve has 4 corners, each one a change in which pair is closest — the colours — so the function is piecewise smooth rather than smooth, and its minimum is not where a derivative vanishes. Refining off the sample grid by golden section moves the answer by 1.4e-5, which is what a corner rather than a smooth minimum looks like.

A gap with corners in it

The signed gap as a function of a mechanism's drive: piecewise smooth, with a corner at every change of closest pair, and a minimum that has to be bracketed rather than differentiated.

The bound that says the sweep missed nothing. Every corner of every body is an affine function of its link's two joints, so its speed is the same combination of the joint velocities the mechanism already solves for — and the largest corner speed anywhere on this drive is V = 1.255 per radian. A distance between point sets is 1-Lipschitz in those points, so from each sample the gap can fall no faster than V: the fine lines are those cones. Where two cones cross is the least the gap can be between the samples, and at 60 samples that is 0.0383 — positive, so nothing was missed. The same bound refuses the twelve-sample sweep of the stud machine, where the bound is -0.281.

A sweep that missed nothing

Why a sampled interference check can miss a collision, and the exact Lipschitz certificate that turns a finite sweep into a statement about a continuum.

crank rocker: the closest pair at one position. The site's standard four-bar: ground 4, crank 1, coupler 3.5, rocker 3. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.5438 here, between coupler · frame. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.

Two bars that have to cross

Why a planar mechanism is a stack of parallel planes: which pairs of links conflict, how much of a drive they overlap for, and what an out-of-plane offset does and does not change.

A colouring is not an assignment. A pin joining links in planes 1 and 3 has to pass through plane 2, and anything in plane 2 whose material covers that pin is pierced by it. That condition involves three links at once where a colouring's conditions involve two, and it is invisible to a colouring because permuting colours preserves a colouring and destroys betweenness. It does not change the number of planes on any machine here. What it removes is arrangements: of the 6 three-plane colourings of a crank rocker, 2 can be built, and of Peaucellier's 192 exactly 96 can.

A plane is a colour

The fewest planes a mechanism can be assembled in, found exactly rather than greedily — and the ordering condition on pins that a colouring cannot express.

The room crank rocker needs. The shaded region is every point any part of this machine occupies at some position of its drive, computed as an occupancy grid over 150 solved configurations and outlined by marching squares — so the outline drawn is the boundary of the set the area was counted from rather than a second object that agrees with it. Its area is 12.78 square units, filling 54% of the box that contains it. The dashed rectangle is the box the joints need, which is what every figure that treats links as lines could have told you; the material needs a box 20% larger in area.

The room a machine sweeps

The swept region of a mechanism: an occupancy grid over solved configurations, its area by lattice and by random points, and the rotating body whose swept disc has an exact area.

The box the joints need, and the box the machine needs. Every machine in the catalogue, measured twice over the same drive: the extent of its joint positions, which is what a site drawing links as lines can report, and the extent of its material, which is what has to fit in something. The ratio runs from 1.13 to 1.36 — between thirteen and 36 per cent more area than the skeleton suggests, on machines whose links are a twentieth of their length wide. The last column is how much of the material's box the swept region actually fills, and it is where the packaging argument really is: a machine at 50% is a machine with a great deal of room inside its own envelope that nothing may be put in.

The hole the machine needs

The footprint of a mechanism: the box its material needs against the box its joints need, and how much of that box the swept region actually fills.

Turning all the way round, against being made of something. Seven four-bars, classified by Grashof's inequality on their four lengths and then asked a question Grashof cannot answer: with a bearing pedestal at each ground pivot, how wide may the links be? The two instruments have nothing in common — one is an inequality on four numbers, the other counts sign changes of (B − A) × (G − A) over a sweep — and they agree about something Grashof was not for. Every four-bar that turns all the way round sweeps a link straight over a ground pivot, so its closest approach is exactly zero and no positive width is admissible; not one of the rockers does, and they take widths up to 0.20 of their shortest link.

The crank that cannot turn all the way

The width a four-bar's links may be given a pedestal at each ground pivot, found by bisection and by counting sign changes — and why full rotation and a bearing at both ends are incompatible.

Free space, in pieces. The driving angle round the circle, with the arcs at which the machine is both assembled and clear drawn heavy. One stud in the way takes a bite out of the turn and leaves 1 arc: the crank can still reach every remaining angle by going the other way. Two studs leave 2, covering 74% of the turn — and every configuration in both arcs is a perfectly good solution of the same constraint equations, on the same assembly branch, at the same mobility. Nothing a solver computes distinguishes an angle in one arc from an angle in the other; what separates them is that the machine cannot be driven from one to the other.

Free space comes in pieces

The set of driving angles at which a machine is both assembled and clear: a union of arcs, bounded by collisions rather than by singularities, and disconnected by two obstacles where one leaves it whole.

A pin is not a point

What a revolute joint costs once it has a size: the boss round it, the spacing two pins on one link need, the column it makes through a layered machine, and where each of those bites.

A link, as a distance and as a body. The same two links every field before this one has drawn as lines, drawn as the material they are made of. A bar is a rectangle with its ends rounded off to the bosses that surround its pins, and it is convex; a bell crank is two arms meeting at a shared pin, and it is not — its inner corner turns the wrong way by 0.52, which is the single fact that puts it outside every separating-axis test here. The pins are marked because they are what has not changed: the constraint equations are the same, the solve is the same, and the positions are the same. What is new is everything between the pins.

A body is all size

Why the bodies field's quantities are all sizes, what a measurement of a mechanism's angles says about them, and the one dimensionless number the field has.

$The gap is a straight line in the width. Three machines, four widths each, every width a fraction of that machine's own limit. Each set of points is collinear to the last bit of a double — the slopes wander by less than 10⁻¹³ across the range — because a feature-to-feature distance is linear in the corners of the two bodies and a bar's corners are linear in its width. The slope reads the contact: Chebyshev's linkage at -2.692, a crank rocker with a post at -1.350, a crank passing a stud at -1.350. A bar's boss grows 1.35 times as fast as its side, so −1.35 is a boss against something that is not growing and −2.70 is boss against boss. Extending each line to zero gives the widest link the machine will take, and the bisection that finds it the hard way agrees — 0.30113 against 0.30112, inside the bisection's own residual.$

The gap is a straight line in the metal

The worst gap over a whole drive measured at several link widths, found affine to fourteen digits, with the slope reading which features are in contact and the widest buildable link predicted from two points.

36% shared, and they never touch. The region left arm visits over a whole drive, the region right arm visits, and — in the third colour — the part of the plane both of them visit. The shared area is 4.652 square units, 35.9 per cent of the smaller of the two regions. The two parts are drawn at the configuration where they come closest, and at that configuration the gap between them is 0.3799 — more than twice a link's width, and positive everywhere else on the drive. A test that asks whether the regions intersect has reported a collision between two parts that are never in the same place at the same time.

The regions overlap and the parts never meet

Two parts' swept regions intersected cell by cell against their pairwise gap through the whole drive, with the conservatism measured and the case where it vanishes identified.

One link, five bodies. Five bars with the same two pins, offset by -0.24, -0.12, 0, 0.12, 0.24 of the link's length, drawn to a common scale. Every one of them holds its two pins exactly the same distance apart, so every one of them is the same link: put any of them into a mechanism and the mechanism solves to the same joint positions at every configuration. Nothing in the kinematics of this collection — no loop equation, no velocity, no coupler curve, no mobility count — can tell them apart. What they do not have in common is which ground they occupy on the way from one pin to the other.

A link may be bent

A bar bent by a stated offset, with the mechanism's joint positions unchanged to the last bit, the clearance swept against the bend, and the one contact no bend can reach.

The drive in 2 arcs. Chebyshev's two arms, whose regions over the whole drive overlap though the parts never come within 0.380 of each other. The outlines are each part's region over the whole drive. The filled ground is what a test that sweeps the drive in 2 equal arcs still flags: the points some arc's sweep of one part shares with the same arc's sweep of the other. With 2 arcs that is 0.859 square units. The parts are drawn at their closest approach.

Four arcs and the false alarm is gone

The swept-region interference test made arc-wise: the drive cut into K equal arcs, each part swept over each arc on a shared grid, and the union of the arcs' shared cells measured against K for Chebyshev's straight-line linkage and a pair from Peaucellier's cell; nested partitions never flagging more ground; a grid-free certificate from the least gap in each arc and the slower part's fastest point speed; and a colliding pair refused at every K.

Every shared cell asks for a cut between two visits; 2 cuts answer all of them. Chebyshev's two arms: each of the 5,510 cells of ground both parts occupy at some point of the drive, as the stretch of drive angle between one part's last visit to it and the other part's first — every such cell is visited by each part in one unbroken stretch, so each gives exactly one interval. Sorted by where they end, one in 21 drawn. A partition of the drive clears the pair exactly when every interval holds a cut. The vertical lines are the fewest cuts that do, placed greedily at the earliest unstabbed end: 1.072 and 1.428, making 3 arcs of 0.360, 0.356, 0.274 radians. The thicker intervals are the ones the rule cut at; they do not overlap, so no partition clears the pair with fewer cuts.

Three arcs, cut in the right places

The arc-wise interference test with its cuts placed rather than spaced: every shared cell's interval between the two parts' visits, the fewest cuts that stab them all found greedily and proved least by disjoint witnesses, the shared ground coloured by the cut that separates it, equal arcs against halving where flagged against the fewest, and what each costs in sweeping.

Four cuts serve the whole cell

The arc-wise region test on whole machines: every pair of parts that never touches but shares whole-drive ground in Chebyshev's, Watt's and Peaucellier's straight-line linkages; each pair's fewest cuts; the machine's fewest cuts for all pairs at once, proved least; the grid test on the machine's partition; and the planes each machine is built in when its conflicts come from a region test over equal arcs or over its own cuts.

3 handovers, and 3 cuts where the shared ground stands empty. Four in turn: each tool's lift against the cam angle, from its own cam, with a thick bar under it while its front is inside the shared ground. The dashed lines are the fewest cuts that clear every pair of tools at once, 3, at 114.5°, 204.5°, 294.5°, and the shaded bands are the windows each may move in, 65° to 115°, 155° to 205°, 245° to 295°: the times when no tool is inside the shared ground. The pairs need 1 cut at worst and 6 between them; the station changes hands 3 times after the turn's seam.

A cam machine is cut at its handovers

An arc-wise region test on a work station served by cam-driven tools: the station in plan with each tool's whole-turn ground and the square they share; the timing diagram for four tools in turn, for one tool that strikes twice, and for two jaws that hold while two tools take turns, each with its fewest cuts and their windows; every pair's intervals and cuts against the machine's; the empty time against the gap between strokes, closed form and grid, to the collision; and the three schedules' cuts, handovers and planes.

Two stations share a cut when they hand over together

A region test on a cell of two work stations, each shared by its own cam-driven tools, with a swinging transfer arm that visits both: the cell in plan; the timing diagrams with the stations out of step and in step, each station's empty times and the fewest cuts; the cell's cuts against the moment one station hands over, with each station's own; and a ledger of both schedules with the seam in the arm's swing and inside a visit.

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