The circle of points going straight

Take a four-bar at one crank angle and ask which points of its coupler plane are, at that instant, travelling in a straight line. Not staying on a straight line — nothing does that except on a very particular linkage — but momentarily having zero curvature in their path.

The answer is a whole circle of them, and the circle passes through the pole. It is worth being clear about how surprising that ought to be. The condition “this point’s path has zero curvature” is one equation in the two coordinates of the point, so the answer being a curve is expected. The answer being a circle is not, and the site’s rule is that a locus asserted to be a circle and a locus measured to be a circle are different objects.

Where the circle comes from #

The path curvature of a point is $\kappa = N/|z'|^3$ where $N = \operatorname{Im}(\overline{z'}z'')$, so the locus of zero curvature is the zero set of $N$. Write $w$ for the point’s offset from the moving frame’s origin in fixed-frame coordinates; then

$$z' = A' + i\varphi'\,w, \qquad z'' = A'' + (i\varphi'' - \varphi'^2)\,w,$$

both affine in $w$. Multiplying out $\overline{z'}z''$ gives four kinds of term — a constant, one in $w$, one in $\bar w$, and one in $|w|^2$ — and the imaginary part of the last one has coefficient

$$\operatorname{Im}\!\left(-i\varphi'(i\varphi'' - \varphi'^2)\right) = \varphi'^3.$$

That is the whole reason the locus is a circle. The quadratic part of $N$ is $\varphi'^3|w|^2$, which is a real multiple of $x^2 + y^2$: equal coefficients on the two squares and no cross term. A conic with equal squares and no cross term is a circle, and completing the square gives the centre and radius in closed form.

It also says exactly when the locus is not a circle. When $\varphi' = 0$ the quadratic part vanishes and what remains is linear: the inflection locus of a plane that is instantaneously translating is a straight line. That is not a degeneracy to be papered over. A plane that translates has no pole — every point has the same velocity, and no point is still — so the circle has nowhere to pass through, and the correct object is a line at infinity’s worth of distance away. The figures refuse to draw a circle there rather than drawing an enormous one.

Measuring rather than asserting #

The derivation above is three lines and is correct. The site’s habit is nonetheless to check it against something that could have come out differently, and here that is easy: sample the zero set and fit a general conic to it.

A general conic is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, six coefficients, normalised to unit length. Nothing about the fit assumes the curve closes, that it is bounded, or that it is round. If the locus were a slightly eccentric ellipse the fit would say so, in the cross term $B$ and in the gap between $A$ and $C$.

Over six positions of the crank the worst value of $|B|/(|A|+|C|)$ is $8.1\times10^{-15}$ and the worst $|A-C|/(|A|+|C|)$ is $6.4\times10^{-15}$. Both at rounding. The locus is a circle.

Three further facts come with it, each computed independently and each required to agree.

The circle passes through the pole, to $8.3\times10^{-12}$ of a coupler length. This one is trivially true from the algebra — the pole has $z' = 0$, so $N = 0$ there — but it is worth checking, because it ties the fitted circle to the pole computed by a different route entirely.

Its diameter is the pole’s own speed divided by the plane’s angular rate. The fitted diameter and $|\dot P|/|\varphi'|$ agree to $1.5\times10^{-16}$ relative. This is the classical $\delta$ from Euler and Savary’s relation, and having it come out of a curve fit is the closest this field gets to a coincidence that is not one.

The diameter is perpendicular to the pole’s velocity. The sine of the angle between them is $1.1\times10^{-16}$. So the circle sits with the pole at one end of a diameter and the pole tangent as the tangent to the circle there, which is the arrangement every textbook picture shows and which is here a measurement.

The polar form, which does all the work #

For computing, the circle is best not held as a centre and a radius. Along a ray from the pole in direction $\psi$, $N$ is a quadratic in the distance $r$ with a root at $r = 0$, because the pole is on the locus. So

$$N(P + r e^{i\psi}) = n_1 r + n_2 r^2,$$

and the second crossing is at $r = -n_1/n_2$ exactly, with $n_2 = \varphi'^3$ known in advance and $n_1$ the directional derivative of $N$ at the pole. One evaluation of a gradient per ray, no root-finding, and no differencing.

That form is $r = \delta\sin\psi$ in disguise, with $\psi$ measured from the pole tangent. It is also what makes the intersection with the cubic of stationary curvature tractable in a later essay, since that curve has the same polar structure with a double root at the pole instead of a single one.

The polar form is also where an early version of one figure went wrong in an instructive way. Sampling the ray at a fixed small step and extracting $n_1$ and $n_2$ by finite differences worked, but the circle is routinely twenty times the size of the linkage, so extracting a coefficient of $r^2$ from samples at $r \approx 10^{-3}$ when the root is at $r \approx 47$ is a cancellation problem, and the fitted circle had a residual of $5.5\times10^{-7}$ rather than $10^{-15}$. Getting the coefficients from the algebra instead removed the step size from the calculation, and with it the only tuning parameter the figure had.

The circle decides where a coupler curve bends #

The circle would be a curiosity if it stayed still. It does not: it is a different circle at every crank angle, and it sweeps across the coupler plane as the mechanism turns. That sweeping is what makes it useful, because it settles a question about whole curves.

A coupler curve has an inflection exactly at the instants when the moving inflection circle crosses the tracing point. The curve’s curvature is zero precisely when the point is on the circle; a sign change in that curvature is an inflection; so the inflections of the curve are the crossings.

That is checkable in two directions and both are checked. At each instant where the exact curvature changes sign, the tracing point’s distance from the inflection circle is measured, and the worst value over the crossings of a coupler curve is $2.5\times10^{-16}$ of a coupler length. And the count itself is taken twice: once from the sign changes of the exact curvature, and once from the turning of the drawn polyline — three consecutive traced points, the sign of the cross product of the two chords, and count the changes. That second route knows nothing about derivatives, poles or circles; the only thing it shares with the first is the solver that placed the points. For the coupler point at $(0.8, 0.2)$ of the coupler both routes report two.

Most coupler points never inflect at all #

Since the circle sweeps only a limited region of the coupler plane, a point outside that region never gets crossed and its curve bends one way for the whole cycle. That suggests a survey, and the survey is more lopsided than expected.

A grid of 1,681 body points over a five-by-five region of the coupler plane, each one’s curve traced through a full turn and its inflections counted: 1,389 of them — 83 per cent — have none at all. 278 have two and 14 have four. Nothing has an odd number, which is a check rather than a finding: a closed curve cannot.

The two moving pins are in the majority, for the plainest reason available. A pin traces an exact circle, a circle has no inflections anywhere, and the inflection circle can sweep over a pin without anything happening — because when it does, the pin’s curvature is zero and also constant, which is a contradiction unless the pin is at the pole. In fact the pin is never on the inflection circle except when it is at the pole, and that is the sense in which the survey’s answer at those two points is not merely typical but forced.

Why a circle, and not something else #

It is worth asking why the answer is a circle rather than, say, an ellipse, because the algebra gives the answer without giving the reason.

The reason is that the quadratic part of $N$ comes from a single product: $\overline{(i\varphi' w)}\cdot(i\varphi''-\varphi'^2)w$, which is $\bar w w$ times a constant. $\bar w w$ is $|w|^2$, and $|w|^2$ is rotationally symmetric. No other pairing of terms in the expansion produces a second-order term at all, because $A'$ and $A''$ contribute no $w$ and the two $w$-linear pieces multiply to nothing of second order after the imaginary part is taken.

So the roundness is inherited from the structure of a rigid motion rather than from anything about four-bars. A rigid motion contributes a rotation to the velocity field and that rotation is $i\varphi'w$ — a rotation, uniform in every direction — and the quadratic term is that piece meeting the corresponding piece of the acceleration field. Any planar rigid motion has that structure, so any planar rigid motion has a circular inflection locus, whatever produced it. A cam follower’s face, a wheel on a road, a sheet of paper being pushed about: all of them, at every instant.

That generality is worth stating because it is the strongest available version of this field’s opening claim. The circle is not a fact about linkages. It is a fact about rigid motions in the plane, and linkages are how this site happens to make them.

The two ways a point can be on it, and only one is interesting #

A point is on the inflection circle if its path curvature is zero. There are two ways for the numerator $N = \operatorname{Im}(\overline{z'}z'')$ to vanish, and they are not the same situation.

The first is that $z''$ is parallel to $z'$: the point is accelerating along its own direction of travel and not across it, so the path is straight and the point is merely speeding up or slowing down. That is the ordinary case, and it is the one every point on the circle is in.

The second is that $z'$ itself is zero, which is the pole. A point that is not moving has no direction of travel, no path tangent, and no curvature; putting it on the circle is a convention that happens to be consistent with the algebra rather than a statement that the pole is going straight. The site’s machinery refuses to report a curvature there, and the circle’s polar form uses the pole’s membership as a computational convenience — it is the known root that makes the quadratic solvable in one step — while the essays are careful not to describe the pole as travelling in a line.

The distinction is not pedantry. It is the reason Ball’s point is defined as the intersection of the circle with the cubic other than the pole, and the reason the search for it explicitly excludes the pole rather than discovering it as one of several answers.

What this says about coupler curves as objects #

The curves field treats a coupler curve as an algebraic object: a sextic, of a certain genus, with a certain number of double points, traced by three different linkages because of Roberts’s theorem. None of that machinery says anything about inflections, because inflections are a real-geometry question rather than an algebraic one — a real sextic can have any number of them up to a bound, and which number a particular one has depends on where the real points fall.

This essay’s route answers it directly and cheaply. It also answers the version of the question a designer actually has, which is not how many inflections does this curve have but does this curve have a nearly straight bit, and where. A nearly straight bit is a stretch where the tracing point is near the inflection circle for a while, which is a statement about the point’s position relative to a moving circle rather than about the curve’s algebra.

That is exactly the reading the classical straight-line linkages get in a later essay. Watt’s linkage keeps its tracing point near the inflection circle over a stroke; Peaucellier’s does something different in kind, which is why it is exact and Watt’s is not; and the point where the linkage would do best at one instant is not the point either designer chose, which is worth a measurement.

The circle at a dead centre, and at a translation #

Two positions deserve saying explicitly, because both are places where a naive implementation returns a number and should not.

At a dead centre the loop’s Jacobian is singular. The derivatives of the coupler angle do not exist — the mechanism’s motion is not differentiable in the input there — so $N$ has no coefficients and there is no circle. The machinery refuses rather than returning a very large one. The site has a whole essay on the two different things called jamming and this is one more consequence of the first of them.

At an instantaneous translation the pole runs off to infinity and, as above, the locus becomes a line. That is a perfectly ordinary configuration for a four-bar — the coupler’s angular rate passes through zero twice a turn on many of them — and it is not a failure of anything. The figures handle it by not drawing a circle, and the inflection survey handles it by counting the sign changes of curvature rather than crossings of a circle, which is why that measurement is stated in terms of curvature and the circle is offered as the explanation rather than as the method.

The distinction matters because it is the difference between two claims that sound the same. The inflections of a coupler curve are the crossings of the inflection circle is true where the circle exists. The inflections of a coupler curve are the sign changes of its path curvature is true everywhere and is what is counted. The first explains the second; it does not replace it.

The two numbers again #

Everything in this essay comes out of the same two numbers the previous one ended with: the direction the pole is travelling and the length $\delta$. The circle is placed by them entirely — one end of a diameter at the pole, the diameter along the pole normal, the length $\delta$.

Which raises the obvious question, and it is the one the field’s remaining essays are about. If two numbers settle the curvature of every point of the plane, what is left over? What is left over is how those curvatures are changing, and that needs a third derivative. It produces a second locus, this one a cubic rather than a circle, and where the two loci meet is the one point of the moving plane whose path is straight and staying straight.

Most coupler points never inflecting is the observation worth ending on, because it says what the inflection circle is actually for. The circle is small — it passes through the pole and its diameter is one length of the motion — and a coupler point sits somewhere on a rigid body that is generally much larger. So over a full revolution the circle sweeps a region, and a point outside that region never lies on the circle at any instant and its path never has an inflection anywhere. Inflections are a property of points near the pole, and the pole is near the mechanism’s own middle. That reframes what a designer is doing when hunting for a coupler curve with a straight stretch in it: not searching a large space for a rare event, but confining the search to the region the inflection circle sweeps — which is computable in advance, from the motion alone, before any coupler point is chosen. That is a genuine reduction of a search space and it comes free with the circle, which is the ordinary payoff of having a locus rather than a test.