Four positions brought together

This site has two pieces of machinery that ought to be the same machinery, written eight days apart for different reasons, and this essay is the measurement that they are.

The first is Burmester’s construction for four prescribed positions of a moving plane. Given four poses, which points of the moving plane have their four images lying on one circle? Such a point can be a crank pin, with the circle’s centre as its fixed pivot, and the locus of them is a cubic — the circle-point curve. That machinery was written for the synthesis field and has been drawing figures since.

The second is the cubic of stationary curvature: the points of a moving plane whose path curvature has momentarily stopped changing.

They have to be the same curve in the limit. A circle through four positions that are coalescing is a circle of four-point contact, and four-point contact with a circle is exactly the condition that the curvature is stationary. So bringing the four poses together should carry the first curve onto the second.

Why the comparison is one-dimensional #

The obvious way to compare two curves is to measure a distance between them — a Hausdorff distance, or the worst gap between corresponding points. Both are bad measurements here and for the same reason: a distance between two drawn curves is a quantity about how they were sampled and where they were clipped.

The cubic of stationary curvature is already in polar form about the pole: one crossing per ray, at $r = -s_2/s_3$, exactly. The finite circle-point curve can be put in the same form by fixing a ray and bisecting the four-position concyclic defect along it. So along one fixed ray each curve has one crossing, and the gap between the two crossings is a length, with no sampling in it and no clipping.

Six spreads, from 0.32 down to 0.01 radians of crank, give six gaps: 0.0897, 0.0205, 0.00505, 0.00126, 0.000315, 0.0000788 coupler lengths. The ratios are close to four throughout, and the fitted slope on log-log axes is 2.02.

The first wrong way to space the four positions #

Getting that number took two attempts and both failures are worth recording, because each of them produced a clean-looking exponent that was wrong for a reason that had nothing to do with the geometry.

The first arrangement was the obvious one: positions at $-1.5$, $-0.5$, $+0.5$ and $+1.5$ spreads, symmetric about the instant. With that spacing, the concyclic defect of a point of the cubic falls as the fifth power of the spread and the defect of an ordinary point falls as the third.

Fifth is wrong. A point of the cubic has four-point contact, so its four coalescing positions should leave a circle at fourth order. What produced the extra order was the symmetry: an even arrangement kills the odd term of the expansion for reasons that are about the arrangement rather than about the point. The spacing was doing the work the geometry was supposed to do.

The second wrong way #

So the spacing was made deliberately unequal: $0$, $1$, $2.3$, $3.7$ spreads. That fixed the exponent of the defect — a point of the cubic came out at 3.96 and an ordinary point near 3 — and broke the curve comparison instead.

The four positions now have their centre at $1.75$ spreads after the instant being measured. So as the spread shrinks, the finite curve converges on the cubic of a different instant, one that is itself moving toward the target at a rate proportional to the spread. The result is a first-order convergence: the fitted exponent came out at 0.80 and the gaps fell by factors of 1.6 to 1.9 rather than by 4.

That failure is more instructive than the first because the number it produced was not obviously wrong. An exponent of one is a perfectly respectable answer for a convergence, and a reader shown 0.80 with six clean points behind it has no reason to doubt it. What gives it away is that the reason for it is nowhere in the geometry: nothing about the relationship between a four-position curve and an infinitesimal one suggests first order.

The arrangement that works is unequal and centred: $-1.75$, $-0.75$, $+0.55$, $+1.95$. No symmetry to cancel a term, and no drift of the point being converged to.

What the four-position construction is for #

Worth restating, because the limit is easier to follow if the finite object’s purpose is in view.

A designer wants a linkage that puts a body through four prescribed poses. Each of the four-bar’s two moving pins has to be a point of the moving body whose four positions lie on a circle, so the two pins must be chosen from the circle-point curve, and their fixed pivots from the corresponding centre-point curve. Four poses is the largest number for which the answer is a curve rather than a finite set: at five poses the count drops to four points, and at three every point of the plane works and the construction is a circumcentre.

So the four-position curve is where a designer has a one-parameter family of choices, and choosing well within it is where the branch, circuit and order defects get decided. It is a working object rather than a theoretical one.

The infinitesimal version answers the corresponding question for a path rather than a set of poses: which points of the moving plane stay closest to a circular arc near this instant. Same question, continuous rather than discrete.

The measurement runs the site’s own machinery #

The comparison could have been made with a re-implementation of the concyclic condition, and it is not.

The finite side calls the synthesis field’s own defect — the same function the defect survey sweeps 1,176 synthesised linkages with, and the same one the four-position figures are drawn from. If that function had a bug, this measurement would show it, which is the point. A comparison between two pieces of machinery is only a check if at least one of them is load-bearing elsewhere.

That has caught something on this site before, in the other direction: the solver’s Jacobian sign error was invisible to every check that used the solver on both sides of a comparison, and became visible only when a quantity computed through it was checked against something that did not go through it at all.

Two exponents, and what they are counting #

The convergence of the two curves is one measurement. There is a second, closer to the geometry, that measures the same thing point by point.

Fix a body point, take four positions of the moving plane spread by $s$ about the instant, and ask how far the four images are from lying on one circle — the concyclic defect. For an ordinary point that defect falls as $s^3$; for a point of the cubic of stationary curvature it falls as $s^4$. Measured, with the centred unequal spacing: 3.22 and 3.96.

Those exponents count the same thing the contact-order exponents count, one order lower, because a defect measured from three points plus a fourth is one order less sensitive than a distance measured from a circle that was fitted exactly. The pairing that matters is that they separate: a point of the cubic is closer to concyclic than its neighbour and stays closer as the positions coalesce, by one whole order.

The ordinary point’s 3.22 is not 3.00 and the gap is honest rather than tidy. The sequence is still drifting toward 3 at the smallest spreads sampled, and pushing further runs into the concyclic defect’s own conditioning: a circle fitted through three points that are a hundredth of a radian apart on a smooth curve is an ill-conditioned object, and the site has met that before from the other side, where a three-point circle fit got worse with finer sampling.

The centre-point curve, and where the pivots go #

The circle-point curve lives in the moving plane; the corresponding centre-point curve lives in the fixed plane and holds the pivots. The two are related by the same conjugate correspondence Euler and Savary’s relation describes, applied to whole curves rather than to single points.

In the limit that correspondence is exactly the ray relation: a point of the cubic of stationary curvature has its osculating circle’s centre on its own ray from the pole, at the distance the relation gives. So the infinitesimal centre-point curve is the image of the cubic under the conjugate map, and it is another cubic.

The site draws the finite version and computes the infinitesimal one, and the two agree in the same limit and by the same argument. Nothing new has to be said about it, which is the sign that the correspondence is doing real work rather than being a relabelling.

A limit with a rate is worth more than a limit #

It would have been possible to say that the finite curve tends to the cubic and leave it there. The statement is true and it is in the literature.

What a rate buys is a falsifiable version. The two curves converge is compatible with a great many implementations, including several wrong ones — a curve that converged to the wrong cubic would still converge. The two curves converge as the square of the spread, and here are six spreads and the fitted slope is not: an implementation converging to a nearby but wrong curve would show a slope that flattens, and one converging to the cubic of a drifting instant shows a slope of one, which is exactly what the second failed attempt above produced.

The site’s rolling field made the same argument in almost the same words about the Lie bracket: an agreement that did not degrade with the step would have meant the two routes were one route. Here the degradation is the evidence.

Why the two machineries were written apart #

There is a fair question about why a site with the four-position construction already in it needed a second, infinitesimal version of the same thing, and the answer is that the two are wanted for different jobs.

The finite construction takes poses as its input: four positions of a body, given. It is the designer’s object. Nothing about it requires a mechanism to exist yet — that is the whole point, since the mechanism is what is being synthesised — and its output is a set of candidate pivots.

The infinitesimal one takes a motion as its input: a mechanism, running, at a position. It is the analyst’s object. Its output is a statement about which points of an existing coupler behave well, and it says nothing about how to build anything.

Those are opposite directions and they meet in the middle, which is what this essay measures. A designer who wanted the infinitesimal object would have to make up four poses to feed the finite one; an analyst who wanted the finite object would have to prescribe poses that the mechanism already passes through. The limit is the statement that neither detour is necessary.

It is also the reason this field is not simply an extension of the synthesis field. Everything in synthesis starts from a wanted motion; everything here starts from a motion that exists. The overlap is one theorem wide.

The exponent, and what a failure would have looked like #

The convergence is quadratic, so at each halving of the spread the gap falls by four. It is worth spelling out what the alternatives would have meant, because the exponent is the only part of this measurement that carries information about correctness.

An exponent of one means the thing being converged to is itself moving at a rate proportional to the spread. That is the second failed spacing above, and it is what a badly centred set of positions produces.

An exponent of zero — a gap that stops falling — means the two curves are simply different curves that happen to be close. That is what a wrong sign or a wrong coefficient in either construction would produce, and it is the failure the essay would most want to catch.

An exponent above two means a cancellation is occurring for reasons outside the geometry, which is the first failed spacing. Tidier-looking and equally wrong.

Two is the answer the geometry predicts: the finite curve’s departure from the infinitesimal one is governed by the first term of the expansion that the four positions do not capture, and with the positions centred that term is quadratic. Getting 2.02 from six spreads is therefore not a decoration on a limit; it is the only part of the measurement that could have come out differently.

The rate turns the limit into a licence #

A limit says the two curves agree eventually. A rate says how nearly they agree at a stated spread, and that turns the whole comparison from a consistency check into something a designer can use.

The gaps fall as the square, and reading a coefficient off them puts the relation at roughly $0.8\,s^2$ — 0.0897 at a spread of 0.32 radians, 0.0205 at 0.16, 0.00505 at 0.08, with successive ratios of about four. So the departure of the four-position curve from the cubic of stationary curvature is about eight tenths of the square of the spread, in the units the polar comparison uses.

Invert it and the useful number appears. A one per cent agreement needs a spread of about $\sqrt{0.01/0.8}$, which is 0.11 radians — a little over six degrees of crank. Four prescribed positions falling inside about six degrees may be synthesised against the cubic instead, with a one per cent error that is stated rather than hoped for, and the cubic is a closed-form curve where the four-position construction is a solve.

That is worth having because the two constructions cost very different amounts. Burmester’s four-position curve is computed by solving a concyclic condition at every candidate point; the cubic is an expression in the motion’s own derivatives at one instant. Where the substitution is legitimate it replaces a family of solves with one evaluation, and the rate says exactly where it is legitimate.

It also says where it is not, which matters more. A designer prescribing four poses across a substantial part of a stroke — thirty degrees, say — is at $0.8 \times 0.27$, a departure of twenty per cent, and the cubic is not an approximation to that problem at all. The infinitesimal machinery is for prescribing a path locally, and using it on a genuinely finite four-position problem is not a small error but a different question answered.

Which is the general shape of what a measured exponent buys over a proved limit. A limit divides the world into two cases and a rate makes it continuous, so the question stops being is this the infinitesimal case and becomes how much does treating it as one cost. That is answerable, it is answerable in the units of the problem, and the answer is a number a specification can carry.

What this does not close #

Two gaps, named rather than implied away.

The five-position limit is not done. Burmester’s construction at five poses gives at most four points rather than a curve, and the corresponding infinitesimal object is the set of points with five-point circle contact — the subject of the next essay. The two ought to be related by the same coalescence argument and this site has not measured that one. Doing it properly needs the four-position defect’s fifth-order analogue, whose conditioning is worse and whose measurement would need care.

The convergence is measured along one ray. The claim is about two curves and the measurement is about two points, one on each. That is a deliberate narrowing for the reason given at the start — a distance between drawn curves measures the drawing — but it does leave open the possibility that the two curves converge along that ray and behave differently elsewhere. The figures overlay the whole curves and they agree visually everywhere; a quantitative statement about the whole curve is not made.