The gap is a straight line in the metal

Every clearance this field has computed was computed at one link width. The width is a design variable — very often the design variable, since a link’s length is fixed by the kinematics and its thickness is not — and nobody has asked what the gap does when it moves.

It does the simplest thing available, and it does it exactly.

The claim, and why it is not obvious #

While one pair of features stays the closest, the worst gap over a whole drive is an affine function of the link width.

It is worth being clear about why that is not free. The gap is a minimum — over every pair of parts, over every feature of each part, and over every angle of the drive. A minimum of many functions is generally not smooth even when each of them is, and this field has already found that the gap is not smooth in the drive angle: it has corners wherever the closest pair changes hands, and its minimum sits on one of those corners more often than on a stationary point.

So the natural expectation is that the gap would be piecewise linear in the width at best, with kinks wherever the argmin jumps. What is measured is that over a useful range there are no kinks at all: the slopes agree to within $2.7\times10^{-14}$, which is the arithmetic’s floor rather than a tolerance.

The reason is that a distance between two features — vertex against vertex, vertex against edge, edge against edge — is a linear function of the corners’ positions, and every corner of a bar is linear in the bar’s width. So as long as the same two features are closest, the gap is exactly affine in the width. The measurement is a statement about how long that condition survives, and the answer is: over the whole useful range.

What the corners are, and why they do not spoil it #

The distinction the previous paragraph turns on is worth drawing out, because this field’s two non-smoothnesses are easily confused.

The gap curve has corners in the drive angle. Those are real and an earlier essay is about them: the minimum over pairs is a minimum of several smooth functions, and a minimum of smooth functions has a corner wherever two of them cross. That is why the worst clearance is usually not where a derivative vanishes.

The question here is different. It is whether the gap has corners in the width, and the answer is that it has the same corners in the same sense — one wherever two pairs swap — but that over the range a designer cares about, no swap happens at the minimum. The swaps happen elsewhere on the curve, where the gap is not critical, and they are visible as the ticks creeping sideways in the figure below.

So the two non-smoothnesses are the same phenomenon seen along two different axes, and the useful fact is that they do not coincide: the place where the curve is roughest in one variable is not the place where it is critical in the other.

The slope reads the contact #

The slope is not a constant and that turns out to be the useful part.

A bar in this field is an octagon rather than a rectangle: its end is a boss that has to surround the pin, its sides are the tangents between the two bosses, and its corners are cut. The boss radius is 1.35 times the half-width, so widening the bar by one unit moves its end outward by 1.35 and its side by 1.00, and its cut corner by something between the two.

Measured, the four-bar with a post reads −1.350 per unit of width, and Chebyshev’s linkage reads −2.692.

• −1.35 is one boss growing against something that is not growing at all. The post is bolted to the frame and has no width parameter, so only one body moves.
• −2.70 would be boss against boss: two bars, both growing at 1.35.
• −2.692 is boss against a cut corner, which grows fractionally slower.

So a number that looks like a nuisance — why 1.35 rather than 1? — turns into an instrument. The slope says what is touching what, without drawing anything, and it distinguishes a contact against a fixed obstacle from a contact between two moving links.

And the worst angle does not move at all #

The stronger half of the measurement is about the argument of the minimum rather than its value.

The four-bar with a post has its worst clearance at 2.11773 radians at every width tested, with a spread of exactly zero. Not “to within the sample spacing” — the same sample, because the curve has been shifted rigidly downward and a rigid shift does not move an argmin.

That is what makes the result useful rather than merely tidy. A designer who has found the critical angle once has found it for every width; the expensive part of a swept clearance check is locating the worst configuration, and the answer does not depend on how much metal the links are made of.

Where it stops: the corners do move #

The rigid-translation claim is conditional, and the condition is visible in the same figure.

The corners of the gap curve are where the closest pair changes hands. Two pairs meeting at a corner have different slopes — one may be boss against post at −1.35 and the other boss against boss at −2.70 — so widening lowers them by different amounts and slides their crossing along the drive. On the four-bar with a post the first corner moves from 3.1285 radians to 3.1722 as the links go from a tenth of a unit wide to 0.22, a drift of 0.044 radians, while the minimum’s own angle has not moved by anything.

That is the honest boundary of the result. The curve is translated rigidly where one pair is critical, and it is reshaped where two pairs trade places. If a machine’s worst clearance sits near a corner rather than in the middle of an arc, widening the links can hand the minimum to the other pair and the argmin will jump — once, discontinuously, at the width where the two lines cross.

Where it stops harder: past zero #

The second boundary is sharper and it is about what the quantity is.

A signed clearance is positive as a distance and negative as a penetration depth, and those are not two branches of one function. The distance between two disjoint convex bodies is attained at a unique pair of points; the penetration depth of two overlapping ones is the smallest translation that separates them, which is attained over a set and is a different minimisation altogether.

The crank passing a stud is the clearest case. Its gap runs 0.0980, 0.0440, −0.0100, −0.0640 at widths 0.08 to 0.20 — the first three on a straight line and the fourth already off it — and then −0.1000, −0.1000, −0.1000, −0.1000 at 0.24 and beyond. The stud is 0.1 across. Once the coupler is wide enough to contain it, the depth to which it is contained stops depending on how much wider the coupler gets, because the shallowest way out is across the stud and the stud has not changed size.

The angle at which the deepest penetration occurs wanders too — 0.245 radians of spread across the saturated range — because a plateau has no unique argmin and the sampling picks whichever sample it likes.

The measurement, and the thing it had to refuse #

Four checks, and the last two are the ones that could have gone wrong quietly.

The slopes must agree. Across three intervals of width, on a machine that stays clear, the slopes are required to differ by less than $10^{-9}$. They differ by $1.8\times10^{-14}$, which is the floor for a difference of two numbers near a tenth.

The angle must not move. Not by less than the sample spacing — by exactly nothing, tested as an equality on the recorded angle. A requirement written with a tolerance here would have passed on a curve whose minimum crept by a sample, and crept is exactly what a nearly-rigid translation would do.

Past zero, the law must fail. This is the negative half and the one this measurement exists to have. The same function is run on a machine that has been widened until it overlaps, and the slope there is required to be zero — the penetration saturated — against a slope of −1.35 while the parts were clear. A requirement that only tested the affine range would have proved that a formula holds where it holds.

And the corners must move. Also a negative half, of the rigid-translation claim rather than of the affine one: the first corner of the curve is required to shift by more than a thousandth of a radian across the widths drawn. It shifts by 0.044. A version of this result that said “the gap curve is translated” without that check would have been stating something stronger than the measurement supports, and the figure would have looked identical.

The first version of the saturation requirement asked for the wrong failure. It required the slope past zero to vary, on the reasoning that a law which stops holding must start wandering. What the penetration depth actually does is stop moving at all — four widths, four identical depths of exactly −0.100 — so the slope was constant, the requirement about variation failed, and the failure was the measurement telling the essay what its own result was.

What this buys, in practice #

The result turns a sweep into a lookup, and it is worth saying exactly how far.

One sweep answers every width. Compute the gap curve at any one width, measure the slope from a second, and every other width is $g(\theta) - c\,(w - w_0)$. The expensive part — a mechanism solved at several hundred configurations, with every pair of bodies tested at each — is paid once.

Two points give the widest buildable link. Extend the line to zero. On Chebyshev’s linkage that gives 0.30113, and the bisection search the field has been using — which rebuilds the machine and re-sweeps it at each of thirteen iterations — gives 0.30112, with a residual gap of $4\times10^{-5}$ at its answer. The two agree to inside the bisection’s own tolerance, from two evaluations rather than thirteen sweeps.

That is a two-routes-to-the-same-number result of the kind these essays collect, and the two routes are genuinely independent: one is a linear extrapolation from the geometry of features, the other is a black-box root find on a function that knows nothing about why it is linear.

One sweep, several tolerances #

There is a second reading of the same law that this field has needed since it started pricing tolerance bands, and it comes free.

A manufacturing tolerance on a link’s thickness is an interval of widths. The affine law says the resulting interval of clearances is the same interval scaled by the slope and shifted — so a ±0.005 tolerance on a bar whose critical contact reads −2.70 per unit costs ±0.0135 of clearance, at the same angle, with no re-solve and no sampling over the tolerance box.

That is a much stronger statement than the usual one. A clearance analysis under tolerance is normally done by sampling: pick a few hundred combinations of dimensions inside the box, sweep each, and take the worst. That is necessary when the dimensions are lengths, because a length changes the mechanism’s motion and every configuration moves. It is not necessary for a width, because a width changes no configuration at all — the mechanism solves to exactly the same angles and the same joint positions, and only the bodies hung on them grow.

That is the real content of the affine law: a width is the one dimension in this field that is not kinematic. Everything else a designer can change moves the machine; this one moves only its metal.

What is assumed, and where it would fail #

Three assumptions, stated because each is a place where the law would stop.

The bodies grow by scaling one parameter. A bar here is generated from its width, so every corner is an affine function of it. A part whose shape changes in a more complicated way — a boss that stays fixed while the web thickens, say — has corners that are still affine in its parameter, so the law survives with a different slope. A part whose outline is remeshed as it thickens does not, and the departure would be at the remeshing’s resolution rather than at the arithmetic’s.

Every part grows by the same amount. If the coupler thickens and the crank does not, different pairs fall at different rates, and the minimum over pairs is then a minimum of lines with different slopes — still piecewise linear, no longer a rigid translation, and its argmin can move at every crossing.

The parts are still clear. Past zero the quantity changes identity, as above.

None of the three is exotic and all three are ordinary design situations, which is why the result is worth stating with its conditions rather than as a rule. What survives all of them is weaker and still useful: the gap is piecewise affine in the widths, with finitely many pieces, and the pieces are separated by contacts changing hands.

What it says about the field’s own habit #

This field has computed a great many clearances at one width — 0.16 of the frame, everywhere, for thirteen essays — and that choice has always looked like an arbitrary convention that every number inherits.

The affine law says it is not arbitrary in the way it looked. A clearance quoted at 0.16 is a clearance quoted at every width, because the conversion is a subtraction with a slope the same computation reports. The convention costs nothing and hides nothing, provided the slope is carried alongside the value — and until now it has not been.

Two numbers rather than one, then, is the shape a clearance result should have in this field: the gap, and what it costs per unit of width. The second is as cheap as the first and it is the one a designer actually acts on, since nobody who finds a clearance of 0.104 wants to know only that it is positive.

The same reasoning applies one level up, to the swept region a machine needs and the box it fits in. Both are computed from bodies, both are quoted at one width, and neither is affine in it — a swept area grows roughly as the perimeter times the offset, so the leading term is linear and the correction is not. That is a different result and it is not measured here; what is worth carrying is that the question is now askable of every quantity in the field, and the answer will be different each time.

Still open: growing a body is not the only way to change it #

Every offset here is uniform: the bar gets thicker everywhere at once. That is one line in a design space with several directions in it, and two of the others are worth naming because this field will need them.

The first is a bar that is not straight. A link is two pin positions and the metal between them is free, so a designer whose coupler fouls a pivot has a cheaper option than making everything thinner. Whether bending buys anything, and against what, is a question the affine law does not answer — a bend is not an offset — and it is what comes next.

The second is an offset that is not uniform along the part: a bar tapered from a wide boss to a narrow middle, which is what a real connecting rod is. The affine law applies to each feature separately, so the gap is affine in whichever parameter controls the feature that happens to be closest — and identifying that feature is what the slope already does.