The second shape is not a choice

Every field on this site so far has been handed the shapes and asked where they go. A four-bar arrives with four lengths and the question is what its coupler draws. A cam arrives with a profile and the question is what the follower does. A gear arrives with an involute flank, and the essay that explains it explains why that curve keeps the ratio constant. The bodies are given; the motion is computed.

This field runs the other way, and it is the last direction left. Put two bodies on fixed centres, turn them in some stated relation, and require them to stay in contact — touching, not interpenetrating, at every instant. Give one of them a shape. The other one’s shape is then not available to be designed. It is determined, completely, by the first shape and the relation between the two motions; there is exactly one of it; and a designer who “chose” a different one would have chosen a pair that does not touch.

One equation, and what is deliberately left out of it #

Let the first body turn about the origin through an angle $\varphi$, and the second about $O_2 = (a, 0)$ through $\psi(\varphi)$. A point of body 1 with body coordinate $p$ sits, in the fixed frame, at $X = R(\varphi)\,p$.

At that place $X$ there are two material points, one belonging to each body, and they are moving differently:

$$v_1 = \omega_1\,\hat z \times X, \qquad v_2 = \omega_2\,\hat z \times (X - O_2),$$

with $\omega_1 = 1$ and $\omega_2 = \psi'(\varphi)$, because every derivative on this site is taken with respect to the input angle rather than to a time nobody stated.

The two bodies are rigid, they are touching, and neither is passing through the other. Whatever else their material points do, they cannot separate or overlap along the common normal — so the component of their relative velocity in that direction vanishes:

$$n \cdot (v_1 - v_2) = 0.$$

That is the whole subject. Everything in this field is that equation with a different profile handed to $n$ and a different pair of motions handed to $v_1$ and $v_2$. It is one scalar equation in one unknown — where along the profile the contact is — so at each instant it is solved by bracketing the sign changes of a dot product and closing each bracket by bisection.

Two details of that solve are worth a sentence, because they are the difference between a figure that can be trusted and one that merely looks right. The first is that the residual is sampled along the whole profile and every sign change is bracketed, rather than a single root being chased from a guess. A profile can carry several contacts at once — a disc riding on a ring of pins touches half of them at every instant — and a solver that returns the first root it finds would report a mechanism with one contact and no way of knowing it had lost the others. The second is that each bracket is closed by bisection rather than by Newton’s method, which is slower and cannot leave the bracket it was given. Near a position where two contacts are about to merge, a Newton step is perfectly capable of stepping across to the other one and reporting a smooth curve assembled out of two different branches.

What is not in the equation is worth as much as what is. The relative velocity $v_1 - v_2$ is the velocity field of a rigid motion, so it vanishes at exactly one point of the plane; that point is the instant centre of the relative motion and the subject calls it the pitch point. Since the field vanishes there, the relative velocity anywhere else is perpendicular to the line joining that place to the pitch point — and the meshing equation therefore says, in other words, that the common normal at the contact passes through the pitch point. That sentence is Willis’s law of gearing, and it is in every textbook on the subject.

It is not what the solve uses. The residual above is formed by taking two rotations, building two velocities, subtracting them and dotting the result with a normal. The pitch point appears nowhere in it. That costs a few extra operations and buys the one thing a check has to have: written the other way round, the check would be asking a formula whether it is itself.

Measured over an external pair of wheels and over a rack cutting a wheel, at every contact the equation returns, the worst distance from the pitch point to the common normal is $2.2\times10^{-14}$ mm. That is arithmetic noise on wheels a hundred millimetres across. The law of gearing arrives here as a measurement, which is the only interesting way for a law to arrive.

It is also worth being clear about what the equation is not asserting. It says nothing about whether the two bodies are touching at some other place as well, and nothing about whether the material is on the correct side of either surface. It is a condition on velocities at a candidate point of contact, and a candidate point that satisfies it can still be inside the other body — which is exactly the situation the undercut essay is about, where a curve is generated correctly and then removed by the same cutter a moment later. Contact is a local statement; what a solid is takes a sweep.

The pitch point is an old friend #

The instant centre of the relative motion of two bodies is exactly the object the linkage field met as the point a mechanism is turning about, and the argument for why it exists is the same argument: a rigid motion has one stationary point unless it is a translation, and a relative rigid motion is still a rigid motion.

Two consequences follow immediately and both matter later.

The first is that for a constant ratio the pitch point does not move. If $\psi' = -k$ with $k$ fixed, the point sits on the line of centres at $a\,k/(1+k)$ from the first centre, whatever $\varphi$ is — and that is the definition of the pitch circles, arriving as a consequence rather than as a construction. The two pitch circles roll on one another without slipping, in exactly the sense the centrodes of a linkage do.

The second is that if the ratio is not constant, the pitch point still exists and simply moves along the line of centres — which is what makes a wheel whose ratio is a function of its angle possible at all, and it makes it without a single new idea.

There is one case with no pitch point, and it is real rather than pathological: if the two bodies have the same angular rate, their relative motion is a translation and there is no stationary point to speak of. The code refuses to return one there instead of reporting a very large number, which is the same discipline the curvature field applies to a radius of curvature quoted at the pole.

The second shape is an envelope #

Solving the meshing equation at one instant gives one point. Doing it at every instant, and asking each time where that point is in the second body’s own frame, gives a curve — and that curve is the second body’s profile.

It has a classical name. The positions of the first profile, seen from the second body, are a one-parameter family of curves; a curve that touches every member of a family and is crossed by none of them is the family’s envelope. Cutting metal does this physically: the material that survives is the material that no position of the cutter reached, and its boundary is the envelope.

The envelope has a second definition, and it is the one worth checking against, because it has no calculus in it at all. Take the family at $\varphi$ and at $\varphi + h$, intersect them, and let $h$ go to zero: the intersection approaches the envelope. Two neighbouring positions of a moving curve cross somewhere, and the place they cross is the place they are about to touch.

Those two definitions have nothing in common as computations. One solves a dot product of a normal with a velocity; the other intersects two polylines and knows nothing about normals, velocities or rigid bodies. So the distance between their answers is a real check, and it should fall as $h^2$ for a symmetric pair of neighbours.

It does: over separations from $0.08$ down to $0.005$ radians the gap falls by a factor of $246$, and the fitted exponent is 1.99.

That measurement took two attempts, and the first one is worth recording because it is a trap this site has walked into before from another direction. At 1,600 points per polyline the gap at the finest separation came out $1.8\times10^{-4}$ against the $1.2\times10^{-4}$ the trend wanted, and the fitted exponent fell to 1.88 — not badly enough to look wrong, just badly enough to be quoted. What that finest step was measuring was the polyline, not the curve. At 4,000 points the same five separations give 1.99. A convergence study has to outrun its own discretisation before it can be read, which is what the curvature field found when a three-point difference stencil reported an error that was entirely its own truncation.

One routine, six mechanisms #

The argument this field exists to make is that the following six things are not six subjects with a family resemblance. They are one computation with different arguments.

A straight rack cutting a wheel. The given shape is a straight line; the relation is a line rolling on a circle. What comes out is the involute — the curve the tooth essay derives from the requirement of constant ratio, arriving here instead as the consequence of dragging a straight edge past a turning blank. Measured against the involute of the base circle it should be: $1.4\times10^{-14}$ mm.

An involute flank against its mate. The given shape is one wheel’s flank; what comes out is the other’s, and it is an involute too.

A cycloidal addendum against its mate. Given an epicycloid on one wheel, out comes the hypocycloid traced by the same rolling circle — to $5\times10^{-17}$ of a radian, which is as close to exactly as sampled arithmetic gets.

A roller follower against a cam. The given shape is a circle carried by a body that slides; the relation is the lift law. What comes out is the cam’s surface, and it agrees with the profile the cams field builds by a completely different construction to $1.6\times10^{-5}$ mm.

A housing against a rotor. Given a two-lobed epitrochoid and a body turning at a third of the shaft’s rate about an orbiting centre, out comes a rotary engine’s rotor, with three corners nobody put there.

A pin against a disc. Given a circle fixed in a housing and a disc on an eccentric, out comes one lobe of a cycloidal drive — and, because the pins are identical and evenly spaced, all the others too.

What being conjugate does and does not promise #

A pair generated this way is conjugate: at every position there is a contact, the ratio is whatever the relation said it would be, and the transmission is exact rather than approximate. That is a strong property and it is easy to over-read, so it is worth saying plainly what it does not include.

It does not promise that the pair can be built. The envelope can double back on itself, in which case the shape it describes is not the boundary of any solid; it can require material where the first body already is; and it can run off the end of the profile it was generated from, which is what the limits of contact are about.

It does not promise that the pair survives being made slightly wrong. Two shapes conjugate at one centre distance need not be conjugate at another, and the difference between the tooth forms that survive that and the ones that do not is the reason essentially every gear since 1900 has been an involute.

And it says nothing about force. The contact is maintained by something — a load, a preload, a spring, the fluid in a chamber — and none of that is here. Every quantity in this field survives with every force unknown, which is the same boundary the site has held since the transmission field: a pump’s displacement is computed here as an area swept per turn, which is geometry, and what a fluid does inside it is somebody else’s subject.

Why the sliding is the interesting part #

One consequence of the pitch point falls out immediately and is worth having early, because it disposes of a common misreading.

The relative velocity at the contact is $\beta$ times the vector from the pitch point to the contact, where $\beta$ is the difference of the two angular rates. So the two surfaces slide at every contact except one: the instant the contact is at the pitch point, where the relative velocity is zero and the surfaces are momentarily rolling.

Measured on a 24 : 36 pair, the sliding speed at the pitch point is $7\times10^{-15}$ mm per radian and rises linearly to nine millimetres per radian at the end of the working arc; the measured speed and the product of the relative rate with the distance from the pitch point agree to $10^{-12}$ throughout. Gears roll at one point of the tooth and slide everywhere else, which is why a used tooth has a band of polish across it rather than a uniform finish, and why “rolling contact” is a description of one instant rather than of a mechanism.

What is not claimed is anything downstream of that sliding. Wear, friction, heat and the film that separates the surfaces all need forces and a material; the geometry says how fast one surface travels over the other and stops there.

What this field will do with it #

The rest of the field is that one equation, pointed at things.

It generates a gear tooth from a straight edge, and then shows the same straight edge generating a different tooth on every wheel it is rolled against — which is why gears are interchangeable, and why the standard for a gear is written as a description of the cutter.

It separates the two classical tooth forms by a measurement neither of them can argue with: what shape the driven wheel would need to be if the shafts were moved a fraction of a millimetre apart.

It takes a demanded ratio that is a function rather than a number and computes the two pitch curves that deliver it — and then shows why almost no demand can be met, which is a closure condition on an integral rather than a matter of drawing skill.

And it puts three mechanisms that look nothing alike — a hobbed gear, a rotary engine and a cycloidal reducer — through the same routine, because a shape that has to stay in contact with another shape has no choices left to make.

The rung’s title is worth pressing on once, because not a choice is a stronger claim than it sounds and it is the reason the field exists. Given the first shape and the relation, the second is unique — not one of several, not a family to be optimised over, but the single envelope the meshing equation returns. So a designer who has chosen one profile has, without noticing, chosen the other; there is no second design decision to make and no room to trade anything against anything. That is unusual on this site, where nearly every design step opens a space rather than closing one. It also says exactly where the design freedom went: all of it is in the first shape and in the relation, and once those are settled the mechanism is determined. Which is why the interesting questions in this field are about the first shape — why an involute, why a circle for a roller, why a pin — and why the second shape is only ever computed, checked, and asked whether it can be made.