The linkage, put back from two curves

This field opened by claiming that a motion is a thing in its own right and the mechanism making it is disposable — and that every planar motion is one curve rolling on another without slipping, so two mechanisms with the same pair of curves make the same motion.

Every essay since has used that claim and none has tested it. This one does.

What has to be shown, and what a picture cannot show #

A figure of two curves touching at a point is not evidence that one is rolling on the other. It is evidence that they touch, which is a great deal less, and the habit these essays are written to says so: a requirement that has never rejected anything proves nothing, and a picture that would look the same if the claim were false is not a test at all.

There are two statements to test and they are independent.

The kinematic one. Given the two curves and nothing else — no link lengths, no loop equation, no mechanism — the moving plane’s position and orientation at every instant should be recoverable. Rolling means the two curves touch at the pole and are tangent there, and a rigid placement is exactly a rotation and a translation, so tangency plus contact is two conditions and a placement has three numbers. The tangent condition fixes the rotation and the contact condition fixes the translation.

The metric one. Rolling without slipping means the two curves lay equal lengths against each other. That is a statement about arc length, it involves no placement and no tangent, and it can be checked without reconstructing anything.

Putting the linkage back #

The recovery is one line. Estimate each curve’s tangent direction at the sample, take the rotation carrying the moving curve’s onto the fixed one’s, and take the translation that puts the two contact points together:

$$e^{i\hat\varphi} = \frac{t_f}{t_m}, \qquad \hat A = f - e^{i\hat\varphi}\,m .$$

Neither side of that has a link length in it. The four-bar’s own placement at the same instant comes from its loop equation, and the two are compared.

That factor is the result, and it is worth being clear about why it matters more than the residual itself.

The only approximation in the recovery is the tangent, estimated by central difference between neighbouring samples, and a central difference is second-order accurate. So a residual caused by the tangent estimate must fall as the square of the sample spacing: halve the spacing and the residual quarters. A residual caused by the two placements genuinely disagreeing would not move at all.

Measured: 4.00. The recovery is exact and the number on the page is the sampling.

Three numbers into two curves, and back #

It is worth counting the information to see that the recovery is not getting something for nothing.

A planar rigid placement is three numbers: two of position and one of angle. A motion is a one-parameter family of them, so it is three functions of one variable. The centrodes are two curves, each a one-parameter family of two numbers — four functions of one variable — but the parameterisation of each is arbitrary, which removes one from each, leaving two.

Two against three, and the recovery works. The missing one is exactly the rolling condition: given the two curves, the requirement that arc matches arc determines how the parameters correspond, and that correspondence is the third function. So the count is three against three and nothing is free.

That also says what a different pair of parameterisations would give. Slide one curve along the other at the wrong rate and the result is a motion with slipping — still three functions, still a perfectly good motion, and not the one the centrodes describe. Rolling is the condition that picks one motion out of that family, and it is why the arc-length measurement below is not a corollary of the recovery but the thing that makes the recovery well-posed.

The arc lengths, which need no placement at all #

The second test is completely independent of the first — it uses neither tangents nor placements — and it is the sharper of the two.

The relative disagreement is 9.0 × 10⁻¹⁰, over a run of 1,372 samples and forty-eight units of arc. A polygonal arc length underestimates a curve’s by a term in the square of the step, so the two estimates are each slightly short — and they are short by the same amount, because a rolling contact makes them the same curve measured twice.

That is the second route, and it agrees. A site that had only the first result would have shown that two curves can be made to reproduce a motion; with the second, the two curves are shown to be in rolling contact in the metric sense as well, which is what the word means.

The residual’s own floor #

The arc-length figure reports a relative disagreement of 9 × 10⁻¹⁰ and it is worth saying what that number is a measurement of, since it is not the accuracy of a geometric claim.

Both arc lengths are polygonal: the sum of straight steps between consecutive samples. A polygon inscribed in a smooth curve is short by a term in the square of the step, so each figure is an underestimate. If the two curves were in rolling contact exactly, the two underestimates would be equal only where the two curves have the same curvature — which they do not, since one is the fixed centrode and the other the moving one, and their curvatures differ everywhere except where the inflection circle passes through the pole.

So the residual is a difference of two truncation errors rather than a single one, and it should fall as the square of the step like each of them. At 2,880 samples it is nine parts in ten billion, which is smaller than the arc lengths themselves by nine orders. What that establishes is not that the identity holds to nine figures — it holds exactly — but that the sampling is fine enough that the identity cannot be hiding a real discrepancy larger than a part in a billion.

That is the honest form of a numerical agreement and it is the form these essays have settled on: not “these agree to nine figures”, but “any disagreement is below the floor the method can see, and here is what sets the floor”.

The curves are not curves that fit on a page #

There is a fact about a four-bar’s centrodes that every picture of them conceals, and it should be said plainly because it changes what the figures in this field are.

The pole of a four-bar is where the crank’s line meets the rocker’s line. Two lines meet at infinity when they are parallel, and the crank and rocker of the four-bar drawn throughout are parallel twice per turn — so the fixed centrode has branches that run off to infinity and come back, and the moving one does too.

The minimum distance is 2.868 and the maximum inside the sampling is 10,565, a factor of three and a half thousand. Every centrode in these figures is drawn inside a stated window, and what is outside the window is not missing detail but a different piece of the same curve, arbitrarily far away.

Two consequences follow and both are used above without being stated.

The arc-length test runs over a piece. Adding arc across a branch that has gone to infinity is adding infinity, so the measurement is taken over the longest contiguous run where the pole stays inside the window — 1,372 of 2,880 samples. The identity is local and that is all it needs to be.

A centrode is not a closed curve a designer could machine. The rolling pair that draws an exact straight line has two circles for centrodes, which is why it can be built as a gear pair; a four-bar’s cannot, which is why four-bars are built out of bars.

What the recovery does and does not say about the mechanism #

The strong reading of this result is that the mechanism is redundant, and it needs one qualification.

What is recovered from the curves is the motion — the placement of the moving plane at every instant, which is everything any other field computes about a coupler. Every coupler curve, every path curvature, every inflection, every Burmester point is a function of the placement and therefore a function of the two centrodes.

What is not recovered is the parameterisation. The curves say which placements the motion passes through and in what order; they do not say at what crank angle, because a crank angle is a fact about a mechanism. So two four-bars with the same centrodes traverse the same sequence of placements at different rates, and every velocity in the field would differ by a factor while every shape would be identical.

That distinction is the same one the scale argument draws in the other direction, and it is worth carrying: the centrodes carry the geometry of a motion and not its timing.

What it costs, against solving the linkage #

There is a practical reading of the result and it goes the wrong way for anybody hoping the centrodes are a shortcut.

Solving a four-bar at one configuration is a closed form: two circle intersections and an arctangent. Recovering the same configuration from the centrodes needs the two curves sampled, which needs the linkage solved at every sample, and then needs two difference quotients and a complex division — so the reconstruction is strictly more expensive than the thing it reconstructs, and it is less accurate by six orders of magnitude.

The centrodes are not a computational device. What they are is a canonical form: the smallest description of a motion that does not mention a mechanism, and therefore the object that says when two mechanisms are doing the same thing. Two four-bars with different lengths and the same centrodes are the same motion; a four-bar and a cognate tracing the same coupler curve are not necessarily the same motion, because a curve is one point’s path and the centrodes are the whole plane’s.

That is what this field’s opening claim was reaching for and it is a stronger statement than the one about curves. Two mechanisms agreeing about one traced point agree about that point. Two mechanisms agreeing about their centrodes agree about every point, at once, for ever.

Why the claim was worth testing rather than quoting #

Three things could have gone wrong, and the third is the one that makes the exercise worth the arithmetic.

The tangent could have been wrong by a sign, and the recovered rotation would then have been the correct one turned through π. That is why the comparison is made between rotations rather than between angles, with both branches allowed and the smaller error taken: a tangent direction has two ends and which one a central difference returns flips wherever the pole reverses along a centrode. Comparing angles would have produced a residual of about 3.14 over half the turn and a great deal of confusion.

The curves could have been sampled at mismatched parameters — the fixed centrode at one crank angle and the moving one at another — and the recovery would then have reported a smooth, plausible, wrong placement. They come from one pass over one angle, which is what keeps them matched.

And the whole thing could have been circular. The pole is computed from the mechanism’s own derivatives, so both centrodes come from the linkage; what is thrown away is not the linkage’s provenance but its equations — the recovery uses the two point sets and a difference quotient, and no loop closure. A stronger version would take two curves from somewhere else entirely and roll them, and that is the next paragraph.

Where this leaves the field’s other results #

Two of the essays before this one rest directly on the claim just measured, and both are strengthened rather than changed.

Cusps are crossings of the moving centrode. The argument is that a point tracing a curve has a corner where it is momentarily still, that the only momentarily-still point is the pole, and that the pole in the coupler’s own plane traces the moving centrode. Every step of that is a statement about the moving centrode as a curve in the body, and the rolling result says that curve carries the whole motion — so the cusp condition is not an incidental property of the pole but a property of the motion’s canonical form.

The exact straight line from two rolling circles is the same statement run backwards. There the two centrodes are known in advance — circles of radius L and 2L — and the motion is defined by rolling them, with the mechanism built afterwards to realise it. That is design from the canonical form, and it is why the straightness is exact with no error term: nothing is being approximated, because the curves are the specification.

The general lesson is that this field has two directions of travel and has been using only one. From a mechanism one computes the centrodes; from a pair of centrodes one gets a motion and then looks for a mechanism. The second is synthesis, it is harder, and it is where the exact results in this field have come from.

What a reader should take from a number like 4.00 #

The convergence factor deserves a paragraph of its own, because it is the shape of evidence these essays reach for most often and it is easy to read as a technicality.

A measured residual on its own says almost nothing. 1.6 × 10⁻⁶ could be a real disagreement between two constructions, or a sampling artefact, or an accumulated rounding error, and the number alone does not distinguish them. What distinguishes them is how the residual moves when the method’s one resolution parameter moves.

A real disagreement does not move at all: the two things being compared differ by what they differ by, and sampling them more finely measures the same difference more precisely. A rounding error moves the wrong way, growing slowly as more arithmetic is done. And a truncation error from a second-order method quarters when the spacing halves, because that is what second order means.

So the useful report is not “the two agree to 1.6 × 10⁻⁶” but “the disagreement is the tangent estimate, and here is the exponent that says so”. The first is a claim about a pair of numbers; the second is a claim about which of three possible causes is operating, and it is checkable.

The same instrument turns up in several fields — the exponent that separates a corner from a smooth minimum, the convergence of the centre-point curve onto the cubic at the square of the spread, the fourth-order contact at a Burmester point — and in every case the exponent is the result and the residual is the instrument’s noise floor. Reading it the other way round is how a measurement comes to be quoted as a precision it does not have.

Still open: rolling two curves that came from nowhere #

The honest limit of the result is that both curves were produced by a four-bar, so the demonstration is that a four-bar can be reconstructed from its own shadow.

The demonstration that would settle it is the other direction: take two curves nobody derived from a mechanism, roll one on the other, and ask what mechanism could produce the result. Any pair of curves of equal arc length rolls; the motion it makes is a well-defined roulette; and whether that motion is achievable by a linkage of a stated kind is a synthesis question with a known answer for some pairs and none for most.

The pair worth trying first is a pair of polygons, because a rolling polygon’s motion is a sequence of rotations about its corners and every quantity in this field becomes a finite calculation. The centrodes are then piecewise constant, the pole jumps rather than sliding, and the inflection circle degenerates — which is either a neat limiting case or a case the field’s apparatus refuses, and finding out which is an essay’s worth of work.

The other thing left open is the inverse motion, and it is nearly free. Swap the two planes — hold the coupler still and let the frame move — and the two centrodes exchange roles: the moving one becomes fixed and the fixed one becomes moving. Every quantity in this field has an inverse-motion counterpart obtained by that swap, and a few of them are famous under other names.

What makes it worth a section rather than a sentence is that the swap is not a symmetry of the mechanism. A four-bar’s frame is a link like any other and inverting it gives a different machine — a crank-rocker becomes a drag-link or a double-rocker depending on which link is held — so the inverse motion of one four-bar is the motion of a different four-bar, and the centrodes say which. That is a correspondence between machines, derived from a correspondence between curves, and the apparatus for both halves is to hand, unjoined.