Every point has a centre

Pick a point on a four-bar’s coupler and let the crank turn. The point draws a curve — a sextic, in general, and a famously awkward one. Near any one instant, though, that curve is doing something simple: it is bending at a definite rate about a definite centre, and if the point were replaced by a crank pivoting at that centre nothing would look different for a while.

There is one such centre for every point of the moving plane. The question this essay is about is whether there is any relation between them, and the answer is that there is exactly one relation, it covers every point at once, and it has two numbers in it.

The centre is on the ray #

The first fact is free and is visible in the figure. A point’s centre of curvature lies on the line joining the point to the pole.

The reason is that the centre of curvature always lies along the path’s normal, and the path’s normal always passes through the pole. That second statement is the pole’s defining property, seen sideways: the velocity of a point is perpendicular to the line from the pole to it, because the velocity field is that of a rotation about the pole. Perpendicular to the velocity is the normal, and the normal is therefore along the ray.

So the whole question is how far along. Write $r$ for the distance from the pole to the moving point and $r_0$ for the distance from the pole to the centre of its path, both signed along the ray. Everything is now one-dimensional.

The relation, and the two numbers in it #

The answer is

$$\left(\frac{1}{r} - \frac{1}{r_0}\right)\sin\psi = \frac{1}{\delta},$$

where $\psi$ is the angle the ray makes with the pole tangent — the direction the pole is itself travelling in — and $\delta$ is a length that depends on the instant but not on the point. This is the Euler–Savary equation. Leonhard Euler had the essentials in 1765 and Félix Savary put it in this form around 1830, well before anybody had the vocabulary of a velocity field.

Two numbers, then: the direction the pole is moving, and one length. Those settle the path curvature of every point of an infinite plane simultaneously. It is worth pausing on how strong that is. There are two degrees of freedom in choosing a point of the plane, and one output — the curvature — so a completely unconstrained answer would be some function of two variables. What the relation says is that this function has a very particular shape and that two numbers pin it down.

Where $\delta$ comes from #

$\delta$ has a meaning that makes the relation less mysterious. Put $r_0 = \infty$ — that is, ask for the points whose path centre is infinitely far away, which are the points whose path is momentarily straight. The relation collapses to

$$r = \delta \sin\psi,$$

which in polar coordinates about the pole is the equation of a circle of diameter $\delta$, passing through the pole, with its diameter along the direction $\psi = 90°$ — perpendicular to the pole tangent. That is the inflection circle and it gets an essay of its own, because the fact that this locus is a circle is not obvious and is measurable rather than assertable.

For now, what matters is that $\delta$ is a length one can point at: the diameter of that circle. It is also, independently, the pole’s own speed divided by the plane’s angular rate, which is the form that makes it computable without ever drawing the circle. Both routes are used here and they are required to agree; over six crank positions they differ in the sixteenth digit.

So Euler–Savary can be read as: every point’s curvature is fixed by where it sits relative to one circle. A point on the circle goes straight. A point twice as far from the pole along the same ray has $1/r$ half as large and therefore $1/r_0$ negative — its centre is on the other side of the pole. A point inside the circle curves one way and a point outside curves the other. The circle is the boundary between the two, and the essay after next is about the fact that a coupler curve’s inflections are exactly the moments when the tracing point crosses it.

Checking it against something that has never heard of it #

The site’s rule is that a classical relation gets computed a second way and the two are made to agree. The second way here is the machinery of the field’s first essay: differentiate the loop equation twice, get $z'$ and $z''$ for the point directly, and form

$$\kappa = \frac{x'y'' - y'x''}{|z'|^3}, \qquad z_{\text{centre}} = z + \frac{i\,z'\,|z'|^2}{x'y'' - y'x''}.$$

That route knows nothing about rays, nothing about $\psi$, nothing about $\delta$, and nothing about the pole. It is a Frenet formula applied to a parameterised curve. The Euler–Savary route measures two distances along a ray and applies the relation. They share no line of arithmetic.

Over twenty-eight comparisons — seven crank positions and four coupler points — the worst relative disagreement between the two centres is $3.5\times10^{-15}$. The worst absolute gap is $4.1\times10^{-15}$ of a unit on a linkage four units across.

Two of the pairs are free, and that is what made it useful #

The construction below depends on a fact that is easy to walk past: a four-bar hands over two conjugate pairs without any work at all.

The crank pin A travels on a circle about the fixed pivot $O_2$. A circle’s centre of curvature is its own centre, everywhere, so A’s conjugate is $O_2$ — exactly, at every position, with no computation. The same for B and $O_4$. Every four-bar therefore arrives with two points of its moving plane whose path centres are marked on the drawing in ink.

That is why the relation was usable on a drawing board. Two known pairs are enough to fix both unknowns in it: $\psi$ is measured from the pole tangent, and both a ray direction and $\delta$ can be recovered from two rays whose $r$ and $r_0$ are known. After that every other point of the plane is a division.

It also explains a feature of the cubic in a later essay that would otherwise look like a coincidence. A point tracing an exact circle has a curvature that is not merely stationary but constant, so both moving pins sit on the locus of stationary curvature at every position of the mechanism, and on the fifth-order locus as well. Two of the four points that essay counts are always hardware.

What the sign of $r_0$ carries #

The relation is written with signed distances along the ray, and the signs carry the interesting cases rather than being bookkeeping.

If $r_0$ and $r$ have the same sign and $|r_0| > |r|$, the centre is beyond the point on the far side from the pole, and the point is going round its centre with the pole between the two. This is the ordinary case for a coupler point near the mechanism.

If $r_0$ is negative while $r$ is positive, the centre is on the opposite side of the pole from the point. That happens for every point outside the inflection circle on that ray, and it is the sign flip that makes an inflection an inflection.

If $r_0$ is infinite, the point is on the inflection circle.

And if $r$ itself is zero the relation is undefined, because the point is the pole and the pole is not moving. That case has to be refused rather than approximated, and it is: asking for the curvature at the pole returns a refusal, not a very large number. A figure that quietly reported an enormous curvature there would carry a caption about an extremely tight bend at the one place in the plane where there is no curve at all.

The classical use of it, which is not this one #

Euler–Savary is presented here as a relation to be checked. That is not what it was for.

It was a construction. A designer with a mechanism on the drawing board knew two conjugate pairs for free: the crank pin turns about the fixed pivot, and the rocker pin turns about the other fixed pivot, so two points of the moving plane have their centres already marked on the drawing. From those, Euler–Savary gives the inflection circle; from the inflection circle it gives the centre of curvature of any third point; and from that it gives whether the coupler curve near that point is nearly straight, nearly circular, or neither.

None of that needs a derivative and none of it needs a number. It is one relation applied three times with dividers and a straight edge, and Bobillier’s construction turns even the arithmetic into two intersecting lines. It is the reason a nineteenth-century draughtsman could reason about coupler curves at all.

The site’s ability to compute the same things by differentiating a loop equation is not an improvement on that so much as a different relationship to it. The construction was a way of getting an answer; the computation is a way of checking the construction, which is only interesting because the construction was already correct.

The relation is about a motion, not a mechanism #

The strongest thing to notice about the relation is what does not appear in it: any reference to the machine.

$\psi$ and $r$ are the point’s polar coordinates about the pole. $\delta$ is a property of the motion at the instant. There is no link length, no crank angle, no count of bars. Two completely different mechanisms whose motions share a pole, a pole tangent and a $\delta$ assign the same curvature to every point of the moving plane, and near that instant their coupler curves are the same curves.

This is testable and it is worth testing, because it is the field’s opening claim made concrete. Take the four-bar and the slider-crank from the field’s ledger. They have different $\delta$ — 10.33 and 22.54 coupler lengths at the sampled position — so their motions are different and every point’s curvature differs. Change either mechanism’s proportions until the two $\delta$ agree at some position, along with the pole and the pole tangent, and every point of the moving plane gets the same curvature from both. The mechanisms would still look nothing like each other.

An aside on parameterisation, since this is where it first bites #

The curvature of a path is a property of the path as a set of points. It does not depend on how quickly the point moves along it. That is why every quantity in this essay is independent of the choice to differentiate with respect to crank angle rather than time.

It is easy to lose that. The formula for $\kappa$ has $z'$ and $z''$ in it, and both of those do depend on the parameterisation: run the crank twice as fast and $z'$ doubles and $z''$ quadruples. The ratio $\left(x'y''-y'x''\right)/|z'|^3$ is arranged so the dependences cancel exactly — the numerator scales as the cube of the rate and so does the denominator — and it is the arrangement that makes the answer geometric.

This matters practically the first time a point’s path is compared against a circle. The obvious way to compare them is to sweep an arc at a rate matched to the point’s speed at the instant and measure the distance between the two moving points. That is what the first version of the contact-order measurement did, and every exponent came out at two. Nothing was wrong with the curvature: a coupler point does not travel at a constant speed, so an arc that does is already wrong at second order along the path, and the mismatch along the path buried the mismatch across it. The fix was to measure the geometric distance from the point to the circle — the set, not the parameterisation — and the exponents came out at 3, 4 and 5.

The same trap is available here and is avoided the same way. Every quantity in this essay is a ratio arranged to be parameterisation-free, and the ones that are not — the acceleration of a point, which the other pole is about — are flagged where they occur.

The relation is exact, and the mechanism is not the reason #

One last thing, because it is the sort of statement that gets softened in the telling.

Euler–Savary is not an approximation. It is not a second-order approximation to the relation between a point and its path centre, valid for small displacements. It is an exact statement about an instant: at this configuration, this point’s path has exactly this curvature, and the relation gives it exactly.

What is second-order is the use commonly made of it — replacing the path by its osculating circle and pretending the substitution lasts. That substitution has an error and the error has an exponent, and pinning that exponent down is what the rest of this field is for. But the circle itself is not approximate. It is the unique circle with three-point contact with the path at that point, the relation names it exactly, and the site’s independent route agrees with the relation to fifteen digits.

The distinction is the same one the site draws between Watt’s straight line and Peaucellier’s: an exact statement about an approximation is a different object from an approximate statement. Euler–Savary is the first kind.

Test it across mechanisms, not across configurations #

The relation is about a motion rather than a mechanism, and the essay says so and says it is testable. The form the test should take is worth naming, because the obvious test is weaker than it looks.

The obvious test is to check Euler–Savary at many configurations of one mechanism. That is what the twenty-eight comparisons do, and it is a good test of the implementation — but it checks the relation and the mechanism together, and a relation that quietly depended on something the four-bar happens to have would pass it at every configuration.

The strong test is to check it across different mechanisms producing the same motion. Two mechanisms with the same pole, the same pole tangent and the same $\delta$ at an instant must give identical curvatures for corresponding points of the moving plane — and if the relation carried any dependence on the linkage that produced the motion, they would not.

This site has exactly such a pair. The elliptic trammel’s motion is produced by two sliders and a rod, and it is produced again by a Cardan gear pair — a wheel of half the radius rolling inside a fixed ring — with nothing in common between the two as mechanisms. Same motion, two constructions, and the relation has to give the same answer for every point of the moving plane at every instant.

That is the test worth running, and it is the one that would catch the failure the weak test cannot. Any construction that happened to use the four-bar’s coupler length, its crank, or the fact that its centrodes are what they are would agree with itself on a four-bar and disagree the moment the same motion arrived from somewhere else.

The general form is worth having beyond this relation. A claim about a motion should be tested on two mechanisms, and a claim about a mechanism on two configurations. Testing the first kind the second way is the commonest way a dependence goes unnoticed, and it is invisible in the residuals — which come back at $10^{-14}$ either way, on a relation that is right and on one that is right only here.

What the relation does not settle #

Euler–Savary gives the curvature of every point’s path. It says nothing about how that curvature is changing, which is a third-derivative question and is where the next two essays go.

The gap matters more than it sounds. A curvature is one number; a designer who wants a linkage whose coupler point stays on a circle for a while, or stays on a line for a while, needs to know how long the agreement lasts, and that is the rate of change of curvature rather than the curvature. The classical answer is a curve — the cubic of stationary curvature — and it needs one derivative more than anything in this essay.

There is a second gap, and it is the reason the two numbers $\delta$ and $\psi$ are not the whole story. The relation holds at one instant. The pole moves, the pole tangent turns, $\delta$ changes, and the inflection circle at the next instant is a different circle in a different place. A statement about a whole stroke is an integral of statements like this one and is not one of them. Where the site makes a claim about a whole stroke — Watt’s nine per cent, Chebyshev’s twelve — it is measured over the stroke and not extrapolated from an instant, and the essay that compares them against the instantaneous answer is largely about how differently those two questions behave.