The shape that does not mind where the shafts are

Two tooth forms have been used in quantity. The cycloidal tooth, whose face is an epicycloid and whose flank is a hypocycloid, ran clocks and mill gearing for two centuries and is still cut for horology. The involute tooth runs everything else. Both are exactly conjugate: hand either pair to the machinery and the ratio comes out constant to the last digit the arithmetic has.

So the choice between them was made on something else, and it is worth finding out what, because the usual explanations — that the involute is simpler, or newer, or easier to draw — are not true. The involute is the harder curve, it was described by Euler in 1760, and the cycloidal system was the one with the drawing instruments.

The question, asked so that it can be measured #

“What happens if the shafts are not quite in the right place” is a vague question, and there are several sharp ones inside it. The one that separates the tooth forms is this:

Given the driving flank and the ratio the pair is supposed to hold, what shape would the driven wheel have to be at this centre distance — and how far is that from the shape it actually has?

That is answerable with the machinery this field already has. Generate the conjugate of the driving flank at the new centre distance, and compare it with the flank the driven wheel was made with. A rigid rotation between them is a fitting angle rather than a disagreement — a wheel can be turned on its shaft — so what is measured is the spread of the angular offset along the curve, converted to a length at the pitch radius.

At the design distance both pairs give essentially zero, which is the check that the comparison is measuring anything at all: the involute pair comes out at $3.0\times10^{-8}$ mm and the cycloidal pair at $2.4\times10^{-9}$, both of them the interpolation error of the comparison rather than a property of the gears.

Then move the shafts.

The cycloidal pair wants a different wheel #

Displace the centres by two hundredths of a millimetre — a fifth of the thickness of a human hair, a tolerance no workshop would apologise for — and the cycloidal pair wants a shape $2.9\times10^{-3}$ mm different from the one it has. At five hundredths of a millimetre it is $6.7\times10^{-3}$; at a quarter of a millimetre, $2.8\times10^{-2}$; at a millimetre, $7.4\times10^{-2}$.

Those are small numbers in absolute terms and enormous in relative ones: six orders of magnitude above the pair’s own noise floor at the first step. The pair has stopped being conjugate, and it stopped at the first perturbation.

The reason is visible in the construction. A cycloidal tooth is generated by a describing circle rolling in the space between the two pitch circles: rolling outside the first it traces the epicycloid that becomes one wheel’s face, rolling inside the second it traces the hypocycloid that becomes the other wheel’s flank, and the two are conjugate because one circle rolled on both. Move the centres apart and the pitch circles are no longer touching. There is no space between them for a circle to roll in. The construction that produced the pair has lost its premise, and the shapes that came out of it are simply two curves that no longer have anything to do with each other.

The contact path measurement is the same fact from another angle. Fit a circle to the contacts of a cycloidal pair and it comes back with radius $24.000000$ mm against a describing circle of $24$; the residual is $1.2\times10^{-9}$ mm, and a straight line misses the same points by $0.11$ mm, eight orders larger. The contact path is the describing circle. Take away the rolling and there is nothing for the contact to travel along.

What the cycloidal pair does instead of transmitting #

There is a second measurement, and it is the one that makes the first concrete.

Ask the two flanks directly where they are: hold the pair at the ratio it was designed for, and solve for the configuration in which the two given curves are tangent. At the design distance that solve converges everywhere along the arc, with a residual of $10^{-10}$ and a transmission error of $3\times10^{-16}$ radians. At five hundredths of a millimetre out it stops converging over most of the arc, and the reason is not numerical: the residual it is failing to drive to zero is a gap, and the gap is about the size of the offset. The flanks are not touching.

That is what losing conjugacy looks like from inside the mechanism. The two curves that were designed to be in continuous contact are now apart over most of the engagement and touching at isolated positions; the driven wheel sits wherever the next tooth pair has caught it, and the output angle jumps between one contact and the next. A pair of gears in that state still turns, still transmits, and still has a mean ratio equal to the tooth counts. What it does not have is a constant one.

The involute pair does not notice #

The same measurement on an involute pair, at the same offsets and then at ten and forty times them:

The number does not move. It is the same noise floor at two millimetres out that it is at zero, and the flank the new distance wants is the flank the wheel has.

Solved directly rather than by comparing shapes, the transmission error tells the same story: over an arc of the input, at centre distances of 0, 0.5 and 2 mm, the departure from the design ratio is $3\times10^{-16}$, $3\times10^{-16}$ and $5\times10^{-16}$ radians. Those are floating-point zeros.

The reason is one sentence, and it is the whole argument for the involute. An involute flank belongs to its base circle, and the base circle is a property of the wheel. The line of action is the common tangent to the two base circles; move the centres and that tangent changes its inclination but remains a common tangent, so the contact still travels along a straight line and the ratio is still the ratio of the base radii — which have not changed, because nothing about either wheel has changed.

The pressure angle rises, the contact moves, the pitch circles are no longer the circles they were, the arc of contact shortens. The ratio does not move at all.

There is a tidy way to say the same thing that makes the asymmetry between the two forms plain. The involute’s construction refers to one circle belonging to one wheel; the cycloid’s refers to a circle rolling between two wheels. A construction that mentions only one body cannot be disturbed by the distance between two bodies. A construction that mentions the relationship between them is at the mercy of it.

Every other property of the involute that gets quoted follows from that. The line of action stays straight because the tangent to two circles is straight whatever the distance. The ratio stays exact because it is a ratio of two radii that belong to the wheels. Even the ability to cut every wheel with one straight rack is the same fact seen from the workshop: a rack is a wheel of infinite radius, and an involute pair works at any centre distance, so it works at that one too.

What does change, and why it is a different problem #

The involute’s indifference is often overstated, so it is worth listing what a wrong centre distance does cost.

Backlash. The wheels are still conjugate on the driving flanks and the gap between the back flanks has opened up. The teeth were sized to fill the space at the design distance; at a larger one they do not, and the play is an allowance rather than an error — real, measurable, and irrelevant to the ratio while the drive is loaded in one direction.

A shorter contact arc. The line of action is longer between the base circles and shorter between the tip circles, so fewer teeth are in contact at once and the contact ratio falls. Push the centres far enough apart and it drops below one, at which point a pair of teeth lets go before the next one picks up and the drive is a series of impacts. That is a real limit and it is nowhere near two millimetres on this pair.

A larger pressure angle, and with it a larger separating force on the bearings — a consequence with a force in it, and therefore outside this site’s boundary, but worth naming so the indifference is not read as universal.

None of those is a departure from the ratio, and that is the point. The involute converts a positional error into backlash and bearing load, which are things a designer can allow for; the cycloid converts it into a transmission error, which is a thing a designer cannot.

The measurement’s own foundations #

It is worth saying what could have gone wrong with this comparison, because two things did before the numbers above were trustworthy.

The first is that a pair can be conjugate for the wrong reason. An involute flank has a mirror image which is also an involute, and a pair built with the mirrored second flank is also exactly conjugate — it is the back flank, the one that drives when the load reverses. That pair passes the transmission-error test at every centre distance, including two millimetres out. It fails a comparison of shapes, because the shape being tested is not the shape the generation produces. A ratio measurement cannot see which side of a tooth it is standing on, which is why the finding above is stated as a comparison of curves and checked with a ratio rather than the other way round.

The second is the noise floor. The comparison reads an angle off a sampled curve by interpolation, and at four hundred sample points that interpolation was itself worth $7\times10^{-6}$ mm — three orders above what the good case can reach, and enough to hide the cycloidal pair’s first two steps entirely. At six thousand points the floor drops to $3\times10^{-8}$, and the six orders of separation in the result are real. A comparison has to be quieter than the effect it is looking for, and the way to know it is is to measure the floor rather than to assume it.

Then why was the cycloid used at all? #

Because for a long time its disadvantages did not bite and its advantages did.

A clock’s centres are not adjustable and are not wrong. Plates are drilled together, bearings are jewelled or bushed, and the load is a few grams. The centre distance stays where the drawing put it for a century, which is exactly the regime where a form that requires the pitch circles to touch is safe.

Cycloidal teeth can be very small in number. A clock pinion has six or eight leaves, and an involute pinion of that size is deep in undercutting territory — seventeen is where the corner starts eating the flank at twenty degrees. A cycloidal pinion has no undercutting problem of that kind, because its flank is not an involute and there is no base circle for the generation to run past.

The sliding is distributed differently. With the face of one wheel driving the flank of the other, contact is mostly on one side of the pitch point, which was worth having when the lubricant was whale oil and the wheel was brass.

And the involute’s advantage needed a machine tool to be worth anything. Its great property — one straight-sided rack cuts every wheel — is worth nothing to a workshop that files teeth with a template and everything to a factory that hobs them. The involute won when gear cutting became a machine operation, and not before.

What the involute gives up #

The indifference is not free, and the price is paid in a place the comparison above does not look: the involute has a base circle, and below it there is no flank at all.

A cycloidal flank is a hypocycloid all the way down to the root, and the construction never runs out. An involute stops at its base circle — the string has fully unwound — and everything below that radius on a real tooth is fillet rather than flank. On a wheel with plenty of teeth the base circle is well below the working part of the tooth and nobody notices. On a small pinion it is not, and the corner of the cutter comes back through the flank to make matters worse.

That is why the two forms’ weaknesses look like opposites. The cycloid is fragile in the assembly and robust in the tooth count; the involute is robust in the assembly and awkward below about seventeen teeth, where it needs a profile shift or a larger pressure angle to survive. A clock wants six-leaf pinions and unadjustable plates. A gearbox wants a bored housing and thirty teeth. Each system was used where its own weakness did not apply.

The winning property was a side effect #

There is a historical reading of the comparison that sharpens the design lesson, and it is that nobody chose the involute for the property that won.

The involute was adopted because of how it is made. A straight-sided rack rolling on a pitch line generates it, so the cutter is a simple shape, one cutter cuts every tooth count of a given module, and hobbing and shaping both fall out of the same construction. That is an enormous manufacturing advantage and it is the argument that was actually being made in the nineteenth century.

The centre-distance indifference is a consequence of the same construction and not a separate virtue. The base circles are set by the module, the tooth count and the cutter’s pressure angle; moving the shafts touches none of them; the strand between the base circles is therefore the same strand and the ratio is the same ratio. Every step of that follows from the flank being generated by a straight rack, which is why it was true before anybody thought to check it.

So the form that won the competition won it on a property that came free with the property it was chosen for. That is worth noticing because it inverts the usual account, in which the involute is presented as having been selected for its insensitivity — a story that gives the nineteenth century credit for a measurement nobody could make and obscures the argument they were actually having.

The general lesson is the more useful half, and it cuts against a natural habit. A candidate chosen for one reason may be far better than its rivals on a criterion nobody evaluated, and the criterion may turn out to matter more than the one it was chosen on. Manufacturability picked the involute; robustness kept it; and a design process that stopped at both forms are exactly conjugate would have found the two indistinguishable.

Which is the practical form of everything in this essay. Where two candidates both meet the requirement exactly, the requirement has stopped being the discriminator and something else has to be measured — and the something else is nearly always how each behaves when it is not exactly what the drawing says. That measurement is cheap, it is available from the same machinery that established the exactness, and in this case it is worth two centuries of industrial practice.

The general shape of the result #

Strip out the gear vocabulary and the finding is about design rather than about teeth.

Two candidate solutions satisfy the stated requirement exactly. Neither is better on the requirement. They differ in how they behave when a quantity nobody put in the requirement — the centre distance — turns out to be wrong by an amount nobody specified.

That difference is worth more than the requirement, and it is invisible to any amount of analysis of the requirement. The only way to find it is to perturb something the specification is silent about and see which solution notices.

This site has met the same shape twice already: a linkage whose output band under tolerances is set by which lengths it is sensitive to, and a mechanism that is exactly right and cannot be built. The involute is the case where the perturbation is one number, the measurement is unambiguous, and the answer decided an industry.