Any shape has a partner

The tooth of a gear is an involute, and the essay that explains why makes the argument from the law of gearing: the common normal at the contact must pass through the pitch point, the involute’s normal is always tangent to its base circle, and the line tangent to both base circles passes through the pitch point — so an involute pair transmits a constant ratio.

That argument is correct and it is easy to read as more than it is. It shows that involutes work. It does not show that anything else fails, and the natural next thought — the involute is the curve that solves the law of gearing — is false in a way worth measuring, because almost every consequence of it that people believe is also false.

A flank made up to be nothing #

The flank on the left of that figure is a rising curve with a nine-cycle wobble laid on it, chosen for no reason except that nobody would draw it. Its polar radius is $48 + 16t + 0.9\sin 9t$ over a third of a radian: no base circle, no rolling construction, no symmetry, no relation to any classical curve.

Hand it to the same routine and ask for the mate. Over 121 sampled positions there is a contact at every one, the worst residual of the meshing equation is $3\times10^{-10}$, and the curve that comes out is the shape on the right — which also has no name, and which would transmit motion from that wobbly flank at exactly 24 : 36 for as long as the two are in contact.

Nothing about that is a numerical accident. The meshing equation is one scalar condition on one unknown, and at a position where a root exists the contact exists; the ratio it delivers is the ratio the relative motion was told to have, because the relation between the two rotations was an input to the calculation rather than an output of it. Any profile whatever has a conjugate, and the conjugate holds the ratio to machine precision. The involute is one member of an infinite family and so is the wobble.

That is worth stating in the strongest form available, because it reorganises the subject. The question a tooth form answers is not can this transmit a constant ratio — everything can. Every real reason for preferring one over another is a property the law of gearing does not mention.

The relation is symmetric #

The cheapest possible check on the whole construction is also the one that says the most about what a conjugate is: generate the mate, then generate the mate of the mate, and see whether the original comes back.

It does. Taking the arbitrary flank above, generating its conjugate, treating that as a given shape and running the machinery the other way returns the starting flank to $2.3\times10^{-5}$ mm over 61 points; for a well-behaved involute flank the same trip closes to $1.5\times10^{-5}$ mm. Both numbers are the accuracy of interpolating a sampled curve rather than of the generation, which is why they are four orders larger than everything else in this field and still convincing.

The reason the round trip closes is that the meshing equation is symmetric in the two bodies. It says a certain dot product vanishes; exchanging body 1 and body 2 exchanges $v_1$ and $v_2$ and flips the sign of the whole expression, which has no effect on where it is zero. So the phrase “the driving profile” is a statement about which shaft has the motor on it and not about the geometry. Neither shape is the parent of the other, and a picture of a rack cutting a wheel could as truthfully be drawn as a wheel cutting a rack.

That symmetry is also why the check is worth running at all. A sign error in a normal, a velocity built from the wrong centre, a frame rotated the wrong way: each of them would produce a smooth, plausible, wrong curve, and the round trip would not close. It is the meshing equation’s version of the thing this site does everywhere — two routes to one number, with the disagreement published.

What the round trip caught #

The check earned its place the first time it was run, and the mistake it found is the kind that survives every other test in this field.

An involute pair needs two flanks, and the second one can be written down two ways: as an involute of the driven wheel’s base circle with the same handedness as the first, or as its mirror image. Both are perfectly good curves. Both, it turns out, are exactly conjugate to the driving flank — the mirror is the back flank, the one that drives when the load reverses, and it holds the same constant ratio for the same reason.

So the pair built with the mirrored flank passed the ratio test. Solving the two profiles for the angle the driven wheel takes up, and comparing that against the ratio the pair was designed for, gave a transmission error of $3\times10^{-16}$ radians — as exact as anything on this site. It also passed at a centre distance two millimetres out, which is the measurement the next essay is about.

What it failed was the round trip. Generating the mate from the driving flank produced a curve whose polar angle ran the other way from the flank that had been written down, and the two disagreed by more than a millimetre where they overlapped. The shape being tested for conjugacy and the shape the generation produced were different flanks of the same tooth, and every test that looked at the ratio was blind to it, because both are conjugate.

The lesson is one this site keeps relearning in new clothing: a quantity that comes out right is not evidence that the object is right. The transmission error was measuring a relation between two angles, and a relation between two angles cannot see which side of a tooth it is standing on.

A conjugate belongs to a pair, not to a shape #

There is a second reading of “conjugate” that is even easier to slip into: that a curve has a mate, the way a bolt has a nut. It does not. The mate depends on the relation between the two motions, and changing the relation changes the shape.

The measurement in that table is worth reading slowly. The driving flank is one fixed curve: the involute of a base circle of 45.105 mm. Against a ratio of 2 : 3 its mate is an involute of 67.658 mm; against 1 : 1 it is an involute of 45.105; against 5 : 4 it is an involute of 36.084. In every case the mate’s base radius comes out at exactly $r_b\,r_2/r_1$, the driving base circle scaled by the ratio of the pitch radii, and the worst departure from that over the whole set is $5.7\times10^{-14}$ mm.

So the honest answer to what mates with this shape is another question: mates with it while doing what? Fix the relation and the mate is unique. Change the relation and the same curve is one half of a different pair.

There is a special case of that worth noticing, because it is the one people generalise from. For a 1 : 1 ratio between equal wheels the mate of an involute is an involute of the same base circle — the two flanks are congruent. That is where the intuition that “the teeth are the same shape” comes from, and it is true for exactly that case. A 24 : 36 pair has two flanks of genuinely different curvature, which is why the two radii of curvature at a contact sum to a constant while each varies fourfold.

Then why is every gear an involute? #

Given that the law of gearing rules nothing out, the question that matters is what does. The rest of this field is mostly an answer to it, and it is worth listing the candidates here because they are the properties a tooth form is actually chosen on.

Whether the pair survives the shafts being in the wrong place. Two shapes conjugate at one centre distance need not be conjugate at another, and gearboxes are built by people with tolerances. The involute is the shape that does not mind, and that single property is very close to the whole reason for its dominance.

Whether one tool can make all of them. An involute of any base circle is generated by a straight-sided rack, so one cutter makes every wheel of a module and any two wheels it cuts will mesh with each other. A tooth form that needed a different curved cutter for every tooth count would be a different industry.

What the contact does as it moves. The contact of an involute pair travels along a fixed straight line; a cycloidal pair’s travels along a circular arc; the wobbly flank above sends it wandering along a curve nobody would want to specify. That path decides where the load is applied, how the sliding varies, and whether the next pair of teeth picks up before this one lets go.

Whether the shape can be made at all. An envelope can double back on itself, and a shape that doubles back is not the boundary of a solid. Undercutting is the everyday case, and it is a limit on the process as much as on the shape.

None of those is in the meshing equation. All of them are the reasons real teeth look the way they do.

What a pair like this is good for #

The freedom is not only a negative result. A conjugate pair with an unusual profile is exactly what several mechanisms are.

A lantern pinion — the old clock and mill drive whose “teeth” are round pins between two discs — is a conjugate pair whose driving profile is a circle. There is nothing improper about it: the pin is a shape, its conjugate exists, and the resulting wheel tooth is a curve related to the epicycloid rather than to the involute. It was used for centuries because a pin is easy to make accurately when a curved flank is not.

The lantern pinion is worth a moment longer, because it is the clearest case of the freedom being used deliberately. A pin is a circular arc, and a circular arc’s conjugate under a constant ratio is a curve related to the epicycloid — offset inward from the path the pin’s centre traces in the wheel’s frame by the pin’s own radius. The wheel that meshes with it therefore has a definite, computable tooth shape, and the pin has none of the difficulties a curved flank has: it can be turned rather than formed, it can be replaced when it wears, and its accuracy is the accuracy of a cylinder and a hole. Everything that is hard about the pair has been moved into the wheel, which is made once.

A cycloidal pair is the classical alternative, and its own essay measures what it buys and what it costs. What matters here is that it is conjugate for the same reason the involute is, not for a different one.

A cam and its follower is a conjugate pair in which the relation is not a ratio at all but a stated lift law, and the driving profile is a circle carried on a slide. The cam’s surface is an envelope exactly as a tooth is.

And a rotor pair — the two-lobed rotor of a rotary engine against its housing, a pump’s inner and outer rotors — is a conjugate pair where nobody thinks of the shapes as teeth at all.

The one thing the equation can refuse #

A conjugate exists at every position where the meshing equation has a root along the given profile. That qualification is not decorative, and it is the seed of everything the field says about limits.

A profile is a finite arc. As the pair turns, the contact travels along it, and eventually it runs off the end — the tooth’s tip, the rack’s corner, the end of the cam’s lobe. At that moment the equation stops having a root on that profile, contact on that pair of flanks ceases, and either another pair has already picked it up or the drive has a gap in it. That is what a contact ratio counts, and it is a property of the pair of arcs rather than of the curves they lie on.

There is a second refusal, and it is sharper. Nothing in the meshing equation asks whether the contact point is on the outside of either body. A generated point can be correct as an envelope and inside the material of the other shape, in which case the pair as drawn cannot be assembled and the material has to come off. When the offending body is the cutter, taking the material off is called undercutting and is a manufacturing fact; when it is the mate, it is called interference and is a design error.

Both are the same statement: the meshing equation is local. It describes contact at a point, and a solid is what is left after a sweep.

Put the simple shape where accuracy is hard #

The lantern pinion is named above as the clearest case of the freedom being used deliberately, and the strategy behind it recurs often enough on this site to be worth stating as one.

A lantern pinion’s teeth are round pins held between two discs. A pin is the easiest accurate shape a workshop can make: turn it, and it is round to whatever the lathe holds; make a hundred, and they are identical because they came off the same setting. The wheel that mates with such a pinion has a complicated flank, and the generation supplies it — but the wheel is a large part, cut once, on a machine, and inspectable.

So the design has spent its conjugate freedom on making the hard member trivial. The pinion is small, its teeth are small, and small accurate features are the expensive kind; the wheel is large, its teeth are large, and large accurate features are ordinary. Put the circle where the accuracy is hard to achieve and let the generation carry the complexity to where it is cheap.

That is the same move in two other mechanisms this site has drawn. A cam and roller follower puts the circle on the follower — a small, hard-worked, replaceable part — and the complicated shape on the cam, which is large and cut once. A cycloidal drive puts circles on its pins and the trochoidal profile on its disc, for the same reason and with the same division of labour.

Three mechanisms, three centuries apart, one strategy: the member that must be small, numerous or replaceable gets the circle; the member that can be large and cut once gets the envelope. None of that is deducible from the meshing equation, which is indifferent between the two members, and all of it is available the moment the equation is known to be indifferent.

Which is the practical payoff of the whole essay. The freedom the law of gearing leaves is not a curiosity about exotic tooth forms; it is a design variable, and the way it is spent is a question about which member is harder to make.

What is left of the classical picture #

The classical development of gear teeth starts from the law of gearing and produces the involute in about a page, and the impression it leaves is that the tooth form was deduced. Read the other way round, what happened is a search: the law of gearing admits everything, several forms were used — cycloidal in clocks, pin teeth in mills, involute in machine tools — and the involute won on properties the law does not mention, over about a century.

The three forms did not compete on transmission. Every one of them holds a constant ratio exactly, and any surviving example of any of them can be measured to confirm it. They competed on what happens when the wheels are not quite where the drawing says, on how many cutters a workshop has to stock, on whether the teeth can be finished after hardening, and on which of them can be inspected with the instruments of the day. The winner is the one whose good properties are insensitive to the things that go wrong.

That is a better story about design than the deduction is, and it generalises: the constraint that everybody quotes is rarely the constraint that decided anything. The mechanisms in this field are worth having precisely because the same equation produces all of them, and what separates them is always a property somebody had to go and measure.