Where a curve has a corner

A coupler curve is usually smooth. Some of them are not: they have cusps, corners where the curve arrives, stops, and leaves in a different direction rather than turning through it. A cycloid — the curve a point on a rolling wheel’s rim traces — has one every revolution, and its corners are the moments the point touches the road.

That example gives the whole answer away, and this essay is about the fact that it generalises exactly.

A corner is where the point stops #

For a smooth parameterised curve the tangent direction is the direction of $z'$. That direction is undefined when $z' = 0$, and a curve whose velocity vanishes can leave in any direction it likes, including the one it came from.

So a cusp needs $z' = 0$: the tracing point must be momentarily still.

Which point of a moving plane is momentarily still? The pole, and only the pole. Therefore:

The cusps of a coupler curve are the instants at which the tracing point is at the pole.

That is the whole content, and everything else in this essay is a consequence of it or a check on it.

Stated in the coupler’s own plane #

The pole is a point of the fixed plane and also a point of the moving plane, and it is a different point of each at each instant. Its path in the fixed plane is the fixed centrode; its path in the moving plane is the moving centrode.

The tracing point is a fixed point of the moving plane. So “the tracing point is at the pole” is a statement entirely inside the coupler’s own plane: it says that the moving centrode passes through the tracing point.

That reformulation is worth the sentence it costs. It means a designer can decide whether a coupler point’s curve has cusps, and how many, by drawing one curve in the coupler’s own frame and asking which points it passes through — with the fixed plane, the ground pivots and the frame length not involved at all. A body point on the moving centrode traces a cusped curve; a body point off it does not.

The check, and the refinement it needed #

The claim is tested by building it backwards. Take a four-bar, pick a crank angle, compute the pole, and read its body coordinate — its position in the coupler’s own frame. That body point is on the moving centrode by construction, so its curve must have a cusp at that crank angle.

Trace the curve, find where the tracing point comes nearest the pole over a full turn, and see whether it is at the expected angle.

The first version of that check reported a closest approach of $2.2\times10^{-2}$, which is not zero, and looked briefly like a refutation. It was the sweep’s resolution: 720 samples over a full turn is a step of $0.0087$ radians, the minimum was being reported at whichever sampled angle happened to be nearest, and $2.2\times10^{-2}$ is about what a smooth minimum looks like when sampled half a step away from its bottom.

A golden-section refinement of each local minimum costs about sixty function evaluations and takes the closest approach to $1.0\times10^{-14}$ at a crank angle of $1.0000000000$ radians — which is the angle the body point was taken from, to ten decimal places.

That is the second time this site has quoted a sweep’s resolution as though it were a measurement. The bicycle-track essay recovered a wheelbase of $1.0517$ m from a 240-point sweep whose step was $11.7$ mm, and the $1.7$ mm of apparent error was entirely the grid; a golden-section refinement costing nine evaluations took it to $1.0503$ against a true $1.0500$. A minimum quoted at the resolution of the sweep that found it is a measurement of the sweep.

The speed collapses, and that is the observable #

The distance being zero is a statement about the construction. The observable consequence is about the speed.

At the refined crank angle the tracing point’s speed is $9.9\times10^{-15}$ per radian of crank. A fifth of a radian later it is $2.87$. The point is not slowing down and speeding up; it is stopping, and it is stopping in the way a curve with a corner requires.

That measurement is the one worth having because it does not depend on the identification of the cusp with the pole. It is a statement about a traced path: at one instant the point is still, at nearby instants it is moving at ordinary speed, and a path that does that has a corner in it whatever the explanation.

The cycloid, which is this in its simplest form #

A wheel rolling on a road is a planar motion whose centrodes are the wheel’s rim (moving) and the road (fixed). A point on the rim is a point of the moving centrode, so its curve is cusped, once per revolution, at the moments it touches the road. That is a cycloid and everybody has seen the corners.

A point inside the rim is off the moving centrode and its curve — a curtate trochoid — is smooth. A point outside the rim, on a flange, is also off the moving centrode, and its curve has a loop rather than a corner. Three cases, decided by whether a body point is on, inside or outside one curve of the moving plane, and the same three cases appear on a coupler.

That is why the general statement is worth having: rolling a wheel is not a special mechanism, it is a planar motion like any other, and the classification its trochoids fall into is the classification every moving plane’s point-paths fall into.

The site’s rolling field uses the same centrode language for a completely different purpose — there the constraint is that the contact point is instantaneously still, which is the same statement read as a restriction rather than as a description. Both readings are correct and they are about different questions: whether a wheel can be driven sideways, and what shape a point on it traces.

Counting them, and why the count is even #

A closed coupler curve traced by a point that lies on the moving centrode has as many cusps as the number of times the pole reaches that point in a full turn of the crank. That count is a property of how the moving centrode is swept rather than of the point.

On a Grashof crank-rocker the pole runs along a bounded arc of its moving centrode and comes back, so a body point on that arc is reached twice per turn — once on the way out and once on the way back — and the curve has two cusps. On a mechanism where the pole runs all the way round a closed moving centrode, a point on it is reached once and the curve has one.

A wheel on a road is the second case: the moving centrode is the rim, the pole runs round it once per revolution, and a rim point is reached once. That is the cycloid’s single cusp per arch.

The distinction is the same one the timing field makes about whether an input’s travel is used once or twice, and it is decided by the same thing: whether the driving member turns fully or rocks. Grashof’s condition, which decides that, therefore decides the cusp count of every point of the moving centrode — a connection between the site’s oldest classification and one of its newest objects, and one that costs nothing to state once both are in view.

Cusps and inflections are decided by different curves #

Two features of a coupler curve, two loci, and they are not the same locus.

A cusp happens where the tracing point meets the moving centrode. That curve lives in the coupler’s plane and is carried with it.

An inflection happens where the tracing point meets the inflection circle. That circle lives in the fixed plane at each instant and sweeps across the coupler as the mechanism turns.

Both are “the tracing point crosses a moving curve”, and the curves are different objects with different sizes and different behaviour. On the site’s four-bar, 83 per cent of coupler points never inflect because the inflection circle sweeps a limited region; the fraction with cusps is smaller still, because the moving centrode is a single curve rather than a region and a body point has to lie exactly on it.

That last point is worth being precise about. Cusps are non-generic. Move the tracing point by any amount and it comes off the moving centrode and the cusp disappears, replaced by a very tight smooth turn or a small loop. So a coupler curve with a cusp is an infinitely thin case, and a mechanism built to have one will not have one, because its lengths have tolerances.

The three cases, and the one nobody draws #

The classification of what happens near a point of the moving centrode is worth setting out, because two of the three cases are familiar and the third is the interesting one.

Off the centrode, on the near side. The point never stops. Its curve turns tightly where the pole passes closest and stays smooth. This is the curtate case and it is what almost every coupler point does.

On the centrode. The point stops. Cusp.

Off the centrode, on the far side. The point does not stop either, but the pole passes between the point and the centre of its turn, so the point runs backwards for a while: the curve makes a small loop. This is the prolate case, and a coupler curve with a loop in it is common — the site draws several — while a coupler curve with a corner is a measure-zero coincidence.

Reading a curve backwards, then: a loop means the tracing point is just outside the moving centrode at that part of the cycle, a tight smooth turn means it is just inside, and a corner means it is exactly on. The three are a continuous family and the corner is the boundary between the other two.

That is also the practical reason a cusp is hard to observe on a real machine. Aiming for one means aiming for a boundary between two behaviours, and every length in a built mechanism is a range rather than a number, so what gets built is a tight turn or a small loop, chosen by which side of the boundary the tolerances fell on.

The four-bar that was used, and why it is not the default one #

The figures in this essay use a four-bar with ground 4, crank 3, coupler 2 and rocker 3 rather than the site’s usual 4, 1, 3.5, 3.

The reason is that the moving centrode has to pass through a part of the coupler plane where a tracing point is plausible. On the site’s default four-bar the coupler’s angular rate stays modest, the pole stays a long way from the coupler, and the moving centrode is a large curve most of which is nowhere near the bar. Choosing a body point on it means choosing a point several coupler lengths away from any of the hardware, which draws badly and reads as a contrivance.

With a longer crank the coupler swings harder, the pole comes closer, and the moving centrode passes through the region a real tracing point could occupy. That is a drawing decision and it is recorded as one; nothing in the argument depends on the proportions.

What the curve does at the corner #

A cusp has a direction, and it is not arbitrary. Since the velocity vanishes, the leading behaviour comes from the acceleration, and the curve leaves along $\pm z''$ — the same line on both sides, which is what makes it a corner rather than a turn.

The direction of $z''$ at the pole is along the pole normal, which is the direction the inflection circle’s diameter lies in. So the corner of a cusped coupler curve points along the pole normal at the instant of the cusp, and it points that way from both sides.

That gives a check with no numbers in it: the corner in the figure at the top of this essay is perpendicular to the pole tangent, and the pole tangent at that instant is available from the ruler construction with two lines. A reader with a straight edge can confirm the direction of the corner without computing anything.

The trammel’s straight line has two of them #

There is a third mechanism on this site whose curve has cusps, and it is the one the reader would least expect, because its curve is a straight line.

The elliptic trammel is a rod with two ends running in perpendicular slots, and its motion is a circle of radius $L/2$ rolling inside a fixed circle of radius $L$. The rolling circle is the moving centrode, and the rod’s two ends are points on its rim — which is to say the rod’s ends lie on the moving centrode, exactly, which is this essay’s condition for a cusp.

So the criterion predicts that each rod end’s path has cusps, and it does. A rim point of the rolling circle traces a diameter of the fixed one: out to one end, back through the middle, out to the other, back again. At each end of that stroke the point reverses, and a reversal at zero speed is exactly a cusp. Two per revolution, at the two extremes of the slot.

The construction confirms it in the strongest way available. A rim point coincides with the contact point of the two circles precisely when it is at the far end of its diameter, and the contact point is the pole — so the tracing point is at the pole at the two instants the criterion names, with no approximation and no sweep needed to find where.

That gives the trammel’s straight line a description it does not usually get. It is not a smooth curve that happens to be straight; it is a degenerate hypocycloid with two cusps, and the two cusps are the ends of the stroke where a slider reverses. The straightness and the cusps come from the same fact — a rim point of a circle rolling inside a circle of twice its radius — and the essay that measures the straightness to zero is measuring one consequence of the same construction this one measures another of.

It also explains a mechanical fact about that mechanism that the straight-line result leaves unexplained. A slider in a slot reverses at each end of its travel, and this is why: the point’s velocity really does go to zero there, not merely change sign smoothly. That is a cusp in the path and a stationary point in the motion, and it is what makes the end of a trammel’s stroke the place where the mechanism is doing the least — which is a dead centre arriving in a mechanism with no dead centre in the usual sense.

Cusps in the mechanisms the site has already built #

Two places where this has already come up under other names.

The Geneva mechanism is designed so that its pin enters the slot tangentially, and the reason is that a non-tangential entry produces a step change in the wheel’s angular velocity. In the language here, the pin’s path relative to the wheel has to arrive along the slot rather than across it, which is a condition on the direction of a velocity at a boundary rather than a cusp — but the failure mode when it is got wrong is the same discontinuity in the derivative that a cusp is.

The cycloidal drive uses a lobed profile that is a trochoid, and whether it has cusps decides whether the profile can be manufactured at all — a cusped profile has an infinitely sharp corner and no cutter reaches it. That is the same condition as a cam profile whose offset turns itself inside out, which is where this field’s cam essay ends up.

So a cusp is not a curiosity of pretty curves. It is the boundary between a profile that can be cut and one that cannot, and it is reached from three directions on this site.