A construction with no arithmetic in it

The machinery of this field so far needs the motion differentiated twice. To place the inflection circle, the pole’s own velocity is needed; to place a point’s centre of curvature, that plus one relation. Neither is available with a ruler.

But it was. Étienne Bobillier died in 1840 and the constructions named after him were in every kinematics course for a century, and they get all of it from lines that are already in a drawing of the mechanism. This essay is about how, and about the two things the site got wrong reconstructing it — both of them errors of assuming the object was simpler than it is.

Two conjugate pairs, free of charge #

A four-bar hands over two conjugate pairs without any work at all. The crank pin A travels on a circle about $O_2$, so the centre of curvature of A’s path is $O_2$ exactly, at every position. The same for B and $O_4$.

So a drawing of the mechanism already contains two points of the moving plane with their path centres marked. Euler and Savary’s relation has two unknowns in it — the direction of the pole tangent and the length $\delta$ — and two known pairs are exactly enough to determine them. The construction is the geometric version of solving those two equations.

The theorem #

Take two conjugate pairs, $(A, A_0)$ and $(B, B_0)$. Join the two moving points: the line $AB$. Join the two centres: the line $A_0B_0$. They meet at a point $Q$, the collineation point, and the line from the pole $P$ through $Q$ is the collineation axis.

Bobillier’s theorem is:

The collineation axis and the pole tangent make equal angles with the rays $PA$ and $PB$, in opposite senses.

Two lines through the pole that are reflections of each other in the bisector of two other lines through the pole. In directed angles modulo $\pi$ the statement is

$$\theta_{\text{axis}} + \theta_{\text{tangent}} = \theta_{PA} + \theta_{PB}.$$

And on a four-bar the axis is free: $AB$ is the coupler and $A_0B_0$ is the frame, so $Q$ is where the coupler line, extended, crosses the ground line. One intersection of two lines that are already drawn. Rearranged, the theorem then hands over the pole tangent, which is the quantity this field otherwise needs a derivative for.

Measured against the pole tangent computed from the derivatives — the direction of $\dot P$ — the constructed one agrees to a sine of $1.3\times10^{-14}$.

The first mistake: comparing a ray with itself #

The site’s first attempt to check the theorem reported a disagreement of $1.3$ radians and looked, for a few minutes, like evidence that the construction had been misremembered.

It had not. The check was comparing the ray $PA$ with the ray $PA_0$ — the ray to a moving point and the ray to its own centre. Those are the same ray. That is precisely what being conjugate means: the centre of curvature lies on the point’s own ray from the pole, which is the first thing the Euler–Savary essay establishes. The check was asking whether a line makes the same angle with itself as some other line does, which is a question with no content, and it answered no.

The pairing the theorem is about is between two different rays, $PA$ and $PB$. The rays to the two moving points, not the ray to a point and the ray to its centre. Once stated that way the residual is $2.5\times10^{-14}$ radians.

It is a small error and it is worth recording because of what it looked like from the inside: a plausible-sounding statement, an implementation that ran, and a number that was decisively wrong rather than marginally wrong. A decisively wrong number is easy to read as the theorem is false when the theorem is old and the implementation is new.

The second mistake: assuming the axis is unique #

The second error survived longer and is more interesting. It was the assumption that a motion has the collineation axis.

It does not. Each pair of conjugate pairs has its own axis. Take the pair $(A, A_0)$ and $(B, B_0)$ and get one line; take $(A, A_0)$ and some third conjugate pair $(C, C_0)$ and get a different line. On the site’s four-bar at one crank angle, ten such pairs give axes spread over eighty degrees of direction.

What is shared is the relation. Every one of those ten axes is isogonal to the same pole tangent in its own two rays. So the theorem is not “there is a special line associated with the motion”; it is “for any two conjugate pairs, this construction returns a line, and that line is determined by the two rays and the pole tangent”. Which is exactly the content that makes it useful: knowing the axis for the mechanism’s own two pairs gives the pole tangent, and the pole tangent then gives every other pair.

The assumption of uniqueness cost a wrong construction. The natural next step after getting the axis is to use it for a third point: join the new point to A, take where that line crosses the axis, join that crossing to $O_2$, and read off the conjugate where it meets the ray. That construction is correct — but the axis it needs is the axis of the pair $(A, C)$, not the axis of the pair $(A, B)$. Using the wrong axis gave a conjugate 0.9 coupler lengths from the right one, which is not near enough to look like rounding and not far enough to look like nonsense.

The inflection circle, with a straight edge and compasses #

The route that does work is shorter than the one that failed, and it is the classical one.

On the ray from the pole through A, whose centre is $O_2$, the inflection circle crosses at a point $J_A$ satisfying

$$\frac{1}{PJ_A} = \frac{1}{PA} - \frac{1}{PO_2}.$$

That is Euler–Savary with the $\sin\psi$ divided out, because $J_A$’s own centre is at infinity and its $1/r_0$ term is zero, and both $J_A$ and $A$ are on the same ray so the same $\psi$ applies to both. The same construction on the ray through B gives $J_B$. The pole is a third point of the circle. Three points determine a circle.

Compared against the circle the derivatives give, the constructed one agrees to $3.2\times10^{-10}$ of a coupler length in centre position and radius together. Nothing in that route differentiates anything: two divisions, two ray crossings, and a circle through three points.

Reading the theorem as a statement about a quadratic #

There is a way of seeing why the theorem has to be true that makes the isogonal condition less arbitrary, and it is worth a paragraph because it also explains why the axis is not unique.

Euler–Savary can be rearranged into a statement about ray directions. Along the ray at angle $\psi$ from the pole tangent, the map from a point’s distance $r$ to its centre’s distance $r_0$ is

$$\frac{1}{r_0} = \frac{1}{r} - \frac{1}{\delta \sin\psi},$$

which is a Möbius map on the reciprocal distance — an inversion, essentially, with a $\psi$-dependent shift. A projective statement about pairs of points on a pencil of lines through $P$ is exactly the kind of thing that produces a collinearity when two such pairs are joined crosswise, and the collineation point $Q$ is that collinearity.

The angle relation then comes out of the $\sin\psi$: the shift depends on the ray, and the dependence is a sine, so two rays’ shifts combine into a condition that is symmetric in the two rays and antisymmetric about the tangent direction. That is an isogonal condition. The tangent is the axis of the symmetry, and swapping which two rays are used changes the answer — which is why the axis moves when the pair does.

None of that is a proof and none of it is needed to use the construction. It is the reason a reader should not expect the axis to be a fixed feature of the motion: the object that is fixed is the pole tangent, and the axis is a shadow of it cast by whichever two rays were chosen.

What a draughtsman actually did with it #

The classical use is worth spelling out as a sequence, because it makes the value of the construction concrete in a way that a statement of the theorem does not.

One. Draw the mechanism at the position of interest. Extend the crank and the rocker; where they cross is the pole. That is Kennedy’s construction and it costs two lines.

Two. Extend the coupler and the frame; where they cross is $Q$. Join $P$ to $Q$: the collineation axis. Two more lines.

Three. Reflect the axis in the bisector of the rays $PA$ and $PB$. That is the pole tangent. A compass bisection and one line.

Four. On the ray through A, divide to find $J_A$; on the ray through B, find $J_B$; draw the circle through $P$, $J_A$ and $J_B$. That is the inflection circle, and every point of the coupler now has its path’s curvature available by one more division along its own ray.

Eight or nine drawn lines, no numbers written down except in step four, and at the end the local shape of the coupler curve at every point of the moving plane. The modern version computes the same things to fifteen digits and cannot be done on the drawing, and the site’s position is that both are worth having: the computation is what makes the claims checkable, and the construction is what made the subject usable for the century before the computation existed.

Why the construction mattered, and to whom #

It is easy to read this as historical decoration on a computation. It is worth resisting that reading, because the construction answered a question that the computation does not obviously answer better.

A designer with a linkage on a board wants to know where on the coupler to put the tracing point so that the path is nearly straight over the working stroke, or nearly circular, or has an inflection where the mechanism reverses. All of those are questions about the local shape of a coupler curve at a chosen instant, and all of them are answered by the inflection circle and the ray relation.

Doing it numerically requires the loop equation, its derivatives, and a machine. Doing it with the construction requires a straight edge, and — this is the part that matters — it can be done on the drawing already on the board, at any position, in under a minute, and repeated at four positions in five. That is a design loop, not a calculation.

The site’s essay on where an optimiser starts makes the same point from the other end: the atlases of coupler curves that were printed and sold existed because the inverse problem is hard and a designer needed somewhere to begin. The Bobillier construction is the local version of the same need, and it is why the classical literature is full of geometry where a modern treatment would have a Jacobian.

What the construction cannot reach #

Two limits, both worth being explicit about.

It is second order and stops there. Everything the construction gives — the pole tangent, $\delta$, the inflection circle, any point’s centre of curvature — is decided by two derivatives of the motion. The rate at which a curvature is changing is a third-derivative question, and no amount of ruler work on two conjugate pairs will produce the cubic of stationary curvature. There are classical constructions for the cubic too, and they need a third conjugate pair or an additional measured quantity; they are not two lines.

It needs the pole, and the pole runs off to infinity whenever the coupler is close to translating. On a drawing that is not fatal — the constructions have projective versions that handle a pole at infinity — but the computation refuses rather than returning a very large number, for the same reason the inflection circle becomes a line there.

The check that would have caught both mistakes at once #

The assertion behind this essay does three things, and the third is the one that matters.

It checks the isogonal relation on fifty pairs of conjugate pairs across five crank positions, not on the mechanism’s own pair alone. That is what makes the uniqueness assumption impossible to hold: fifty pairs produce fifty axes, and if the code were quietly using one axis for all of them the residuals would be enormous for forty-eight of them.

It checks that those axes are actually different, by measuring the spread of their directions and requiring it to exceed twenty degrees. Without that, a table of fifty rows all agreeing to $10^{-14}$ is compatible with the table being one measurement printed fifty times — which is the shape of circular check the site has been caught by before, in the conjugate-action test that parameterised both flanks by the same variable and announced the ratio was constant.

And it checks the two products of the construction — the pole tangent and the inflection circle — against their computed counterparts, which is the part that would fail if the relation were right and the construction implemented from it were wrong. The first version of this essay’s code passed the angle check and failed that one, which is how the borrowed-axis error surfaced.

Two error sources, and only one of them is the theorem’s #

The construction agrees with the derivatives to $2\times10^{-13}$, and it is worth being clear about what that number is a measurement of, because it is not the accuracy of the method as anybody ever used it.

The theorem is exact. It contributes no error at all: the isogonal relation holds identically, the division that locates $J_A$ is an identity, and a construction executed perfectly returns the pole tangent and the inflection circle to whatever precision the execution has. The $10^{-13}$ is the arithmetic of executing it in floating point, and it says the site’s implementation is not introducing anything.

A draughtsman’s execution had a completely different error, and it was not small. Every step is a drawn line, and a drawn line has a width, a pencil has a point, a compass has a pivot hole, and the intersections of nearly-parallel lines are located badly. The collineation axis is found by intersecting the coupler with the frame — two lines that on many four-bars meet at a shallow angle, which is the worst case for locating a crossing.

So the classical method’s accuracy is a property of the board, not of the theorem, and the two are worth keeping apart because they behave differently. Executing the construction more carefully improves the answer without limit; the theorem cannot be improved, because it contributes nothing to improve. That is an unusual arrangement — most approximations have an error that no amount of care removes.

Which is exactly why the numerical check is worth having rather than being a formality. It separates the two sources: agreement to $10^{-13}$ says the theorem is right and the implementation is right, so any disagreement a draughtsman finds is the drawing. Before the check, a construction that disagreed with a mechanism might have been the theorem, the implementation, or the pencil, and there was no way to tell which.

And it says what a modern user gets that a draughtsman did not. Not a better theorem — the same one — but the theorem’s own exactness with the board’s error removed, which is the whole of the improvement. The construction is one of the rare pieces of classical machinery that loses nothing at all in being computed, because everything it lost was in the drawing.

One line, two centuries #

The relation itself is worth restating in its plainest form, because it is unusually clean for something with two people’s names attached to different parts of it.

At any instant of any planar motion there is a point that is not moving and a direction it is moving in. Every point’s path curvature is decided by its position relative to those two things, through one relation. The relation can be evaluated with a division or drawn with a straight edge. It has no linkage in it, no lengths, no crank angle, and it applies unchanged to a four-bar’s coupler, a cam follower, a wheel on a road and a sheet of paper on a table.

Everything in this essay is a consequence of that, and everything in the next two essays is what happens when one more derivative is added to it.