Six things a shape is not

Every field on this site collects the things that are said confidently and are not true, and this one has an unusual crop, because its objects are shapes. A shape can be drawn wrongly and still look like itself, and a picture is a poor place to hide a millimetre.

Six claims, each with the measurement that settles it.

A note on how these were settled, since the whole point is that none of them is settled by argument. Each claim above is about a shape, and every shape in this field is generated — computed as the envelope of a given curve under a stated motion — so the way to test a claim about one is to generate the shape and measure the quantity the claim is about. That is different from checking a formula, and it is why the numbers below are lengths and widths rather than agreements between expressions.

One: a rotary engine’s rotor is a Reuleaux triangle #

It is not, and this is the most widely repeated of the six.

A Reuleaux triangle is the curve of constant width made of three circular arcs, each centred on the opposite corner. A rotary engine’s rotor is the envelope of its housing — the shape that stays in contact with a two-lobed epitrochoid while turning at a third of the shaft’s rate about an orbiting centre. There is no reason a curve produced that way would be an arc of anything.

The measurement. Take the Reuleaux arcs through the rotor’s own three apexes. The generated flank departs from them by up to 1.21 mm on a rotor of generating radius 100 — and the flank is the one that is right, because a Reuleaux flank would foul the wall.

The measurement that cannot be argued with. A Reuleaux triangle has the same width in every direction; that is what constant width means and it is the only interesting thing about the curve. The rotor’s width runs from 150.0 to 174.1 over the directions, a spread of sixteen per cent. Whatever it is, it is not a curve of constant width.

Where the myth comes from. Both shapes are three-cornered, roughly triangular and symmetric under a third of a turn, and the Reuleaux triangle is the famous three-cornered curve with a rolling story attached to it. There is also a real relationship: as the ratio $K = R/e$ grows, the housing’s lobes flatten and the rotor’s flanks approach the Reuleaux arcs. The engine with no difference between them is the engine with no chambers, so the approximation gets better exactly as the machine gets more useless.

What the difference is for. The flank is shallower than the arc because it has to clear the wall as the rotor swings past the waist of the housing. A rotor built to the arc would touch the housing along its flank at the position where the two curves are furthest apart — which is a mechanism that jams rather than one that seals.

Two: the two gears’ teeth are the same shape #

They are the same shape when the wheels are the same size, and not otherwise.

The mate of an involute flank is an involute of the other wheel’s base circle, and the base circles are in the ratio of the tooth counts. On a 24 : 36 pair the base radii are 45.105 and 67.658 mm, so the two flanks are involutes of circles half as different again in size: they are visibly different curves with different curvature, and only their being generated by the same rack makes them look related.

The measurement. Generate the mate of one fixed flank against four ratios at one centre distance. The mate’s base circle comes out at 67.658, 56.382, 45.105 and 36.084 mm respectively, each matching $r_b\,r_2/r_1$ to $5.7\times10^{-14}$ mm. A conjugate belongs to a pair, and asking what shape mates with a given one is only a question once the relation is stated.

The 1 : 1 case is where the intuition comes from, and it is genuine: equal wheels have congruent flanks. Everything else is a family resemblance.

A second measurement of the same thing. The two flanks at a contact have different radii of curvature, and the teeth field measures them summing to a constant — 41.042 mm on its pair — while each varies by a factor of four along the mesh. Two curves whose curvatures differ fourfold at the same contact are not the same shape by any reading.

Three: the fillet at the root of a tooth is a circular arc #

It is a trochoid: the path of the cutter’s tip corner in the wheel’s frame.

That is not a technicality. A corner is a single point, and the envelope of a family of points is the family itself, so the fillet is the corner’s trajectory — a curve whose shape depends on the cutter’s proportions and on the rolling radius, and which is different at every tooth count.

Two things follow that a drawn arc gets wrong. The curvature at the root — which is where a tooth breaks — is a property of that trochoid, and an arc through the same endpoints has a different one. And the junction between the fillet and the flank is at a definite radius, which is where the usable involute stops; a drawing that runs the involute down to the root circle shows profile that does not exist, since there is no involute below the base circle.

Why arcs get drawn. A fillet arc is what a draughtsman can construct and what a CAD system offers, and the error it makes is small in the middle of the fillet and largest exactly where the fillet meets the flank — which is the one place its shape is load-bearing. It is also the place where the difference is invisible at any sensible drawing scale: on the fourteen-tooth wheel above, the departure between the true trochoid and a plausible arc through its ends is a few hundredths of a millimetre on a wheel 56 mm across.

Four: gears roll on each other #

The pitch circles roll. The teeth slide, everywhere except at one instant.

The measurement. The sliding speed is zero at the pitch point — $7\times10^{-15}$ mm per radian — and rises linearly to about 12 mm per radian at the ends of the working arc, which is between a quarter and a third of the speed at which the contact itself travels. It is not a correction; it is one of the largest velocities in the mechanism.

The confusion is a real object being described with the wrong body’s name. The pitch circles are centrodes, and centrodes roll — that is what centrodes do. But the pitch circles are not made of anything, and the surfaces that are made of something are somewhere else at every instant but one.

What the sliding does to the two wheels differently. Divide the sliding by each surface’s own speed and the result is the specific sliding, which is not symmetric: on this pair it peaks at 1.31 on the 24-tooth wheel and 1.04 on the 36-tooth one. The small wheel of a pair has the harder time of it, and the geometry says so before any material is named. That asymmetry is also why a used pinion shows a wider band of wear than its wheel, and why the polished band across a used tooth sits at the pitch line: the sliding is smallest there and nowhere else.

Five: if a pair is conjugate, it can be built #

Conjugacy is a statement about an instant. Whether a pair of shapes exists as a pair of solids is a statement about an interval, and the two come apart in the most ordinary case there is.

The measurement. On a twelve-tooth wheel cut by a standard rack, the generation produces a flank every point of which satisfies the meshing equation and lies on the involute to $10^{-14}$ mm. Sweep the cutter over that flank at every other position of the cut and 47 of the 240 generated points are inside it — they were made and then removed, the deepest by five hundredths of a millimetre.

The same failure has three other names on this site — a cam profile that cannot be cut, a linkage that assembles at every position and cannot move between them, a synthesis that hits every prescribed point in the wrong order — and it is one failure: local correctness is not global existence.

Six: the involute is the shape the law of gearing requires #

The law of gearing requires nothing of the kind. It says the common normal at the contact must pass through the pitch point, and every profile has a mate satisfying it.

The measurement. Hand the machinery a rising curve with a nine-cycle wobble on it — a shape with no base circle, no rolling construction and no name. Over 121 positions there is a contact at every one, the worst residual of the meshing equation is $3\times10^{-10}$, and generating back the other way returns the original flank to $2.3\times10^{-5}$ mm.

So the involute was not deduced from the law of gearing; it was selected, over about a century, on properties the law does not mention — that it ignores the centre distance, that one straight-sided tool cuts every wheel, that its contact path is a fixed straight line. The deduction in the textbooks is a reconstruction after the fact, and it is misleading in the way that matters: it suggests the question was what shape can transmit motion, when the question was which of the shapes that can, survives being made and assembled badly.

The sharp version. The threshold for that removal is not a matter of judgement: it is where the cutter’s tip line crosses the line of action exactly at the foot of the perpendicular from the wheel’s centre, and bisecting on that difference gives 17.097264 teeth against the classical $2h/\sin^2\alpha = 17.097264$. Two routes, ten figures, and the measurement never sees the formula.

Why shape claims survive so long #

It is worth asking why these six lasted, when a wrong ratio is caught by the first person to measure a shaft.

A shape is checked by looking, and looking is a low-resolution instrument. The rotor’s departure from a Reuleaux triangle is a hundredth of its size; the fillet’s departure from an arc is smaller; two involutes of different base circles look like the same curve at any scale that fits both on a page. Every one of these errors is invisible in the picture that would be used to communicate it, which means the picture cannot correct it and can only propagate it.

The second reason is that a wrong shape does not always fail. A gear drawn with an arc fillet still meshes — the fillet is not a working surface — so the error is repeated by everybody who copies the drawing and is punished only when somebody computes a root stress. A rotor drawn as a Reuleaux triangle is never built from the drawing, because the manufacturer generates the profile from the housing’s equation; the wrong shape lives entirely in the explanations.

That is a general hazard for a subject whose objects are shapes, and it is why this field computes rather than draws. A figure that came out of a solve can be wrong; a figure that came out of a drawing can be wrong and consistent with every other drawing.

What the six have in common #

Five of the six are the same mistake in different clothes: a shape described by the construction somebody could draw rather than by the process that produces it.

A Reuleaux triangle is drawable with a compass and an envelope is not. An arc is drawable and a trochoid is not. Rolling circles are drawable and a sliding contact is not. Two identical teeth are drawable and a pair of involutes of different base circles is a nuisance. In each case the drawable object is close enough to survive a picture and wrong by an amount a measurement finds immediately.

The sixth is different and is the most interesting: a claim that a requirement determined an answer, when it only permitted it. That mistake has no picture in it at all. It comes from reading a derivation backwards, and the cure is the one this field has applied throughout — take the requirement literally, ask what else satisfies it, and find out what actually did the choosing.

Three more that are simply true #

For balance, three claims of the same kind that survive measurement, because a collection of errors is easy to over-read as a warning about the whole subject.

“The line of action is straight for involute gears.” It is, and it is measured rather than constructed: fitting a line to the contacts of a rack cutting a wheel leaves a worst departure of $1.1\times10^{-14}$ mm, and no circle fits them at all.

“One rack cuts every wheel of a module.” It does: five wheels from 12 to 60 teeth, one cutter, every generated point on the involute of its own base circle to $1.4\times10^{-14}$ mm.

“An involute pair is insensitive to centre distance.” It is, to the limit of what can be measured: the shape it wants at two millimetres out is the shape it has, to $3\times10^{-8}$ mm, which is the comparison’s own noise floor.

The pattern is not that received wisdom is unreliable. It is that the claims which survive are the ones somebody once measured, and the ones that do not are the ones that were only ever drawn.

Specify the condition, not the construction #

Five of the six are a shape described by the construction that draws it rather than by the condition that generates it, and the remedy follows directly — it is worth stating, because it turns a diagnosis into a rule for writing a specification.

Describe a shape by what it must satisfy. Conjugate to this flank at this ratio. The envelope of this cutter under this motion. The locus of contact for this housing and this eccentric. Each of those is a condition, each is checkable on any candidate shape, and each determines the shape without anybody drawing it.

Do not describe it by how it is drawn. Three arcs centred on the opposite corners. A circular fillet blended at the root. An arc of radius $r$. Each of those is a recipe, and a recipe has no condition attached — a shape either was produced that way or was not, and there is nothing to measure on the result.

That difference decides whether a claim can be wrong in a way anybody notices. A shape specified by a condition can be tested: generate it, measure the condition, and report the residual — which is what every measurement in this essay does and why every one of the six fell. A shape specified by a construction can only be compared against the construction, so a wrong construction and a wrong shape are the same object and nothing separates them.

It also explains why the constructions are the ones that survived in circulation. A recipe is transmissible: it can be drawn on a board, followed by hand, and taught in a sentence. A condition needs a computation before it produces a picture, which for most of the history of these subjects meant it stayed in the derivation and the recipe went on the drawing.

So the rule for a modern specification is short and the modern situation is what makes it available. State the condition, generate the shape, and keep the residual — and where a construction is wanted anyway, for a drawing or for a cutter path, quote how far it departs from the generated shape. That number is the one every claim in this essay turned on, and it is the one a recipe alone can never carry.

A seventh, which is nearly true #

One claim deserves a separate hearing, because it is repeated as often as the six and is right in a way that matters.

“A cam is a gear with only one tooth.” As a slogan it is loose, and as geometry it is close to exact: a cam and its follower are a conjugate pair whose relation is a stated function rather than a constant, generated by the same routine, with a pitch point that moves and a pressure angle that is the angle to it. This field’s own cross-check generates a cam surface as an envelope and finds it agreeing with the cams field’s offset construction to $1.6\times10^{-5}$ mm.

What the slogan gets wrong is only the word tooth, which suggests the cam repeats. The honest version is that a cam is a conjugate pair with an arbitrary relation, and a gear is a conjugate pair whose relation is a constant and whose profile repeats — and the second is the special case.