How long a pivot stands in for a linkage

This field opened with a claim: a mechanism is one of many ways of producing a motion, and near an instant the motion can be described without it. Everything since has been machinery for that description — a pole, a circle, a cubic, a handful of special points.

This essay is the measurement that makes the claim falsifiable, and it is deliberately the bluntest thing in the field. Throw the linkage away. Replace a coupler point by a single crank pivoting at the centre of its path’s curvature, with the radius that path has there. Drive the four-bar away from the instant, and measure how far the coupler point gets from the fixed circle the crank sweeps.

The answer is a power law, and the power counts how many derivatives of the motion the substitution got right.

What is being measured, exactly #

Take a body point $\zeta$ of the coupler. At the instant, compute its path’s curvature and its centre — from the derivatives, which is the only place a derivative enters this whole essay.

Then fix that circle. Drive the mechanism to $\theta_0 + \Delta$ for a range of $\Delta$, solve, find where the body point has got to, and measure

$$\big|\,\lVert z(\theta_0+\Delta) - c\rVert - \rho\,\big|,$$

the distance from the moving point to the fixed circle. Plot against $\Delta$ on log-log axes and fit a slope.

That quantity is geometric. It is a distance from a point to a set. It does not care how the point is parameterised, how fast the crank is turning, or whether the crank is turning at all — which is the property that makes it the right measurement, and which the first version of it did not have.

The first version returned two for everything #

The obvious way to do the comparison is to build a crank and drive it. Give the crank the angular rate that matches the coupler point’s speed at the instant, sweep both mechanisms by $\Delta$, and measure the distance between the two moving points.

Every exponent came out at two, for every kind of point, and the separation the field’s argument rests on was invisible.

Nothing was wrong with the curvature. A coupler point does not travel at a constant speed, and a crank driven at a constant rate does, so the two points disagree along the path at second order before they disagree across it at third. The tangential mismatch swamped the normal one, which is the only one contact order is about.

Fixing it meant giving up the idea of a substitute mechanism and measuring against a substitute curve. Contact order is a statement about two curves as sets of points, so the measurement has to be about the sets. Once it was, the exponents came out at 3.00, 3.94 and 4.97.

That failure is the field’s own parameterisation warning arriving in the one place the field could not have ignored it. Every quantity in the opening essay is arranged to be independent of how the input is parameterised, and the first measurement of the field’s central claim was not.

The three exponents #

At a crank angle of 1.6 radians on the site’s running four-bar:

The tidiest reading is that the exponent counts agreements. An osculating circle is defined to agree with a path in position, tangent and curvature — three conditions — so the first place it can disagree is the third derivative, and the separation grows like $\Delta^3$. A point of the cubic has $\kappa' = 0$, so the third derivative agrees too and the disagreement moves to the fourth. A Burmester point has $\kappa'' = 0$ as well, and it moves to the fifth.

The coefficients are of ordinary size, so the exponents are not being read off a plot with a tiny signal in it. At a step of 0.2 radians the three separations differ by about a factor of five each; at 0.02 they differ by fifty.

The same measurement against a line #

There is a companion measurement where the substitute is a straight line rather than a circle, and it is what Ball’s point is defined by.

Take the point’s tangent line at the instant and measure the perpendicular distance to it as the mechanism is driven away. An ordinary point’s distance grows as the square, because it has nonzero curvature. A point on the inflection circle has zero curvature, so it grows as the cube. Ball’s point has zero curvature and zero $\kappa'$, so it grows as the fourth power.

Measured: 3.95 for Ball’s point and 1.87 for a point on the same ray at 60 per cent of the distance.

The line version is one order behind the circle version throughout, because a line is a circle with one fewer parameter and so agrees with a curve to one order less at each stage. That relationship is a check in itself: if the two measurements did not stay one apart, one of them would be measuring something other than contact.

What a wrong derivative would look like #

The value of an exponent, rather than a residual, is that it fails informatively.

Suppose the second derivative of the motion were wrong — a sign error, or a missing term. Then the computed curvature would be wrong, the osculating circle would have the wrong radius, and the substitution would disagree with the path at first order. The fitted exponent would be 1, not 3, and it would be 1 for every point.

Suppose the third derivative were wrong. The curvature would still be right, so an ordinary point would still give 3. But the cubic of stationary curvature would be the wrong curve, so a point taken from it would not have $\kappa' = 0$, and its exponent would come back at 3 instead of 4. The separation would vanish while the first number stayed correct, which is exactly the failure a residual-based check would miss.

Suppose the fourth derivative were wrong. The first two exponents survive and the Burmester one drops from 5 to 4.

So the three exponents together are a graded test of the whole derivative stack, and each order of the stack is checked by exactly one of them. That is a better arrangement than three separate residual checks, because a residual check on a derivative needs something to compare against and the exponents compare against traced positions.

Nothing in the measurement uses a derivative #

Worth stating plainly, because it is the property that makes the whole thing worth the trouble.

The measurement traces positions with the site’s Newton solver — the same solver that draws every figure on the site — computes Euclidean distances from those positions to a fixed circle, and fits a straight line through logarithms. There is no differentiation anywhere in it.

The derivatives enter once, at the start, to say where the circle is. Everything after that is arithmetic on solved positions. So a bug in the derivative machinery cannot hide inside the measurement; it can only place the circle wrongly, and a wrongly placed circle shows up as a wrong exponent.

That is the same arrangement the site uses for its most load-bearing checks elsewhere. The Jacobian sign error survived every check that used the Jacobian on both sides of a comparison and was invisible for months; what would have caught it is a comparison against something that does not go through it, which is what this is.

The step range, and why it is a decade #

The fit runs over steps from 0.02 to 0.2 radians of crank — a factor of ten, nine points, logarithmically spaced. Both ends of that range are chosen and both choices can go wrong.

Too small a step and the separation falls below the noise floor. At a step of $10^{-3}$ radians a fifth-order separation is of order $10^{-15}$ times its coefficient, which is where the solver’s own residual lives, and the fitted slope flattens toward zero as the measurement runs out of signal.

Too large a step and the power law stops being the leading behaviour. The next term in the expansion catches up, the log-log plot bends, and the fitted slope comes out between two integers with no indication that it is a blend.

A decade is enough range to distinguish exponents that differ by one — three points per decade would do, and nine gives room to see a bend if there is one — and it sits high enough above the noise floor for a fifth-order measurement. For the Burmester points, whose separation is smallest, the range is narrowed to 0.01–0.12 for exactly that reason and the fitted values stay at 4.96 to 5.03.

The site has been caught at both ends of this before. The rolling field’s wiggle measurement fitted an exponent of 1.997 and reported the relative departure from the leading term as $\varepsilon/2$ across the whole range — an agreement that did not degrade with the step would have meant the two routes were one route. And a three-point circle fit that gets worse with finer sampling is the other end of the same story.

Three exponents and one mechanism, or one exponent and three mechanisms #

There are two ways to read the measurement and they are worth separating.

Read as three points on one mechanism, it says that the coupler plane has structure: most of its points are alike, a curve of them is better, and a pair of them is better still. That is the reading the field is built on and it is a statement about the moving plane.

Read as one point under three descriptions, it says something about approximation. An osculating circle is the best circular approximation to a curve at a point; whether that approximation is good for three, four or five orders depends on the curve, not on the circle. The exponent is a property of the path, and the loci are the places where the path happens to be exceptional.

The second reading is the one that transfers. Nothing in the measurement is about four-bars: given any curve and any point on it, the same procedure returns the order to which the osculating circle fits. What the field contributes is knowing in advance which points of a moving plane will return which answer, without tracing anything.

That is the difference between a measurement and a theory. The measurement takes a point and returns an exponent; the theory takes a motion and returns the set of points that will give each exponent. The site does both and requires them to agree, which is the arrangement every essay in this field is built on.

What happens off the instant #

One more caveat, and it is the one that keeps the exponents from being over-read as a design rule.

The cubic and the Burmester points are computed at one instant. Drive the mechanism a few degrees and they are somewhere else. A point chosen from the cubic at $\theta_0$ is an ordinary point at $\theta_0 + 5°$, and its curve’s curvature — stationary at $\theta_0$ — is changing again.

So the fourth-order agreement is a property of a neighbourhood of one crank angle and not of a stroke. Over a stroke the special point’s error is smaller near the middle and not obviously smaller at the ends, which is exactly the shape the straight-line comparison shows for the line version.

The design version of the same caution: a designer who wants a coupler point that stays close to a circular arc over a stroke is asking an approximation question over a range, and should be fitting rather than matching derivatives. What the loci give is the best starting point for that fit and a guarantee about one instant of it.

The claim, restated with the number in it #

The field opened by saying a mechanism can be thrown away. Here is the version with a rate attached.

A four-bar’s coupler point can be replaced by a single crank, and the replacement is right to third order. Two moving links become one; four joints become two; and the resulting point follows the original to within a distance that falls as the cube of how far the mechanism is driven from the matching instant. At a step of one degree that is a factor of $5\times10^{-6}$ of whatever the coefficient is.

Choose the coupler point from the cubic and the replacement is right to fourth order. Same one crank, same two joints, an error one power smaller.

Choose it from the pair of Burmester points, when they exist, and it is right to fifth.

None of that says a crank is as good as a four-bar. Over a whole stroke the coefficients matter more than the exponents and a crank’s path is a circle while a coupler point’s is a sextic. What it says is that near an instant the mechanism contributes nothing that a circle does not, and how far “near” extends is a number.

Where the coefficient beats the exponent #

The caveat is important enough to have its own section, because the exponents are seductive.

An exponent describes how an error scales, not how large it is. Two points with exponents 3 and 4 have errors in a ratio of $C_3\Delta^3$ to $C_4\Delta^4$, and which is smaller depends on $\Delta$ and on the two coefficients. Below $\Delta = C_3/C_4$ the fourth-order point wins; above it, the third-order one can be better.

That crossing is not hypothetical. Chebyshev’s linkage with Ball’s point has an order of line contact of 3.99 against the coupler midpoint’s 2.01, and over the whole stroke its worst departure is only 1.34 times better — because the stroke is long enough that the coefficient dominates, and because Ball’s point for that linkage traces a much longer stroke to begin with.

So the honest summary is: the exponents settle what happens near the instant and settle nothing about a stroke. Every claim in this field about a stroke — Watt’s nine per cent, Chebyshev’s twelve, Ball’s 3.8 — is measured over the stroke by tracing it, and none of them is extrapolated from an exponent.

What the field ends up having said #

Five essays of machinery and one measurement, and the measurement is the part that could have failed.

A planar motion near an instant is described by a pole, a pole tangent and a short list of derivatives. From those follow a circle of points going straight, a cubic of points whose curvature is standing still, one point whose path is straight to fourth order, and up to two points whose path holds a circle to fifth. Every one of those objects is a locus in the moving plane, none of them mentions the mechanism, and each of them is exactly the set of points where a simpler substitute lasts one order longer than usual.

The exponents 3, 4 and 5 are the reason to care. Without them the two special curves are classical names for level sets of expressions, and with them they are the answer to a design question — where does the pen go so that a circle will do.

The three exponents are the field’s whole content and there is one more thing to say about why an exponent rather than a distance is the right thing to report. A distance depends on the mechanism, its size, its units and where along its motion the measurement was taken; an exponent depends on none of those. Three, four and five are the same three numbers on a four-bar with links of a millimetre and on one with links of a metre, on this site’s mechanism and on any other, and they will be the same numbers for anybody who repeats the measurement. That is what makes them transferable in the way a residual is not, and it is why the field’s central claims are stated as orders rather than as errors. The coefficient is the quantity that carries the size, and it is worth quoting beside the exponent for exactly the reason the two are different: the exponent says which kind of point this is, and the coefficient says how much that is worth on this mechanism. One is a classification and the other is a measurement, and a claim with only one of them is half a claim.