Why a tooth is an involute

Two shafts, turning. A tooth on one pushes a tooth on the other. As they turn the contact point slides along both flanks, so the geometry at the contact is changing continuously.

The requirement is that the speed ratio does not change while all that happens. Not approximately — a ratio that varied through each tooth’s engagement would produce vibration at tooth-passing frequency, which is exactly the noise a badly cut gear makes.

That requirement has a name and a precise geometric form. It is the law of gearing: at every instant, the common normal at the contact point must pass through a fixed point on the line of centres.

Why the normal, and why a fixed point #

The two bodies are rigid and stay in contact, so their velocities at the contact point must be equal along the common normal — otherwise they would separate or interpenetrate.

Work that condition through and it becomes: the ratio of the angular velocities equals the ratio of the perpendicular distances from the two centres to the common normal. For the ratio to be constant, those two distances must be constant, which means the common normal must be a fixed line as the gears turn.

The point where that fixed line crosses the line of centres is the pitch point, and it divides the centre distance in the inverse ratio of the speeds. Two imaginary circles through it — the pitch circles — roll on each other without slipping, and the gears behave exactly as if they were those two smooth wheels.

The involute delivers it #

Unwind a string from a circle of radius $r_b$ and the end traces an involute. Two properties follow immediately from the construction:

• the string is always tangent to the base circle
• the string is always perpendicular to the traced curve

Put those together: the normal to an involute at any point is a tangent to its base circle.

Now mesh two involutes. The common normal at their contact must be tangent to both base circles — and there is only one such line between two circles on a given side. It does not move as the gears turn.

So the law of gearing is satisfied automatically, at every contact position, for any centre distance. That last part is the involute’s decisive practical advantage: pull the shafts apart slightly and the pitch circles change, the pressure angle changes, and the ratio does not, because the base radii have not changed. A cycloidal tooth loses conjugate action if the centre distance is wrong; an involute does not care.

Testing it on the drawn curve #

The claim above is a theorem, and this site’s habit is to check the implementation against it rather than to restate it.

An earlier version of that check was circular and is worth recording. It parameterised both flanks by the involute roll angle, differentiated, and announced that the ratio was constant to the last bit — which it was, because both angles were linear in the same parameter by construction. The calculation restated the definition rather than testing anything.

The honest version takes the flank points the generator actually produces, estimates the tangent by finite difference between neighbouring generated points, takes the normal, and asks whether it is tangent to the base circle. That is a property of the drawn curve rather than of the formula that made it, so an error in the generator shows up.

Measured on a 20-tooth gear it comes out at 2.3 × 10⁻⁵ — and a flank built from circular arcs instead, which is what a hand-drawn or CAD-approximated tooth often is, comes out at 7.6 × 10⁻². The tolerance sits between them with a factor of ten on one side and 380 on the other.

Getting that measurement right took a correction of its own: the distance from the centre to the normal line is a cross product with the normal’s direction, and using the dot product instead measures the distance to the tangent line, which for an involute grows as $r_b t$. The first version did that and reported a 94% error on a perfectly correct flank.

The pressure angle #

The angle between that fixed line and the common tangent to the pitch circles is the pressure angle, and it is the one free parameter of a standard tooth.

It sets the base radius, since $r_b = r\cos\alpha$. Twenty degrees is the modern default; 14½° was standard earlier and gives a smaller base circle and a slenderer tooth; 25° gives a stubbier, stronger tooth at the cost of higher bearing loads, because the force between teeth acts along the line of action and a steeper line pushes the shafts apart harder.

It also decides how few teeth a gear can have before the cutter undercuts it.

Below the base circle there is nothing #

The involute begins at the base circle. Inside it, there is no involute — the string has fully unwound.

For most gears the root of the tooth is below the base circle, and what is there in a real gear is a fillet left by the cutter, not a continuation of the flank. Drawing the involute inward past its base circle is a common and invisible error, and this site asserts that no generated flank point lies inside it.

What this buys #

A tooth form that keeps the ratio exactly constant, tolerates centre-distance error, can be cut with a straight-sided rack cutter — because the involute’s conjugate is a straight line — and meshes correctly with any other gear of the same module and pressure angle regardless of tooth count.

That last property is why gears are interchangeable, and it is not shared by the alternatives. It is the reason essentially every power-transmitting gear made since the nineteenth century is an involute.

The cycloidal alternative, and why it lost #

The involute is not the only curve satisfying the law of gearing. Cycloidal teeth — an epicycloid above the pitch circle, a hypocycloid below — were standard in clockwork and in early machine gearing, and they conjugate correctly too.

They have real advantages. The contact is convex-against-concave below the pitch line, so the contact stress is lower for the same load. Undercutting does not occur in the same way, so cycloidal pinions can have very few teeth — six or seven is normal in a clock, where an involute pinion would need seventeen.

They lost anyway, and the reason is the one property the involute has that no other conjugate curve does: the centre distance can vary without affecting the ratio.

Move an involute pair apart and the pressure angle increases, the contact ratio falls, and the velocity ratio stays exactly what the tooth counts say. Move a cycloidal pair apart and the pitch circles no longer touch, the generating circles no longer match, and the conjugate condition fails — the ratio fluctuates within each tooth.

That difference decides the manufacturing question. An involute gearbox tolerates bearing wear, thermal growth, housing tolerances and assembly slop. A cycloidal one requires the centre distance to be right and to stay right. In a clock, in a brass plate, with jewelled pivots and no load to speak of, that is achievable. In a gearbox it is not.

The second involute advantage is generation. An involute rack has straight flanks, so a cutting tool for involute gears of a given module and pressure angle is a straight-sided rack — one tool cuts every tooth count. Cycloidal teeth need a different cutter for each count, because the generating circle relates to the mating gear.

So the involute won on tolerance and tooling, not on contact mechanics. The cycloid survives where those pressures do not apply: clock trains, some pumps, and the cycloidal-drive reducers whose lobes are cycloids for a different reason.

What “conjugate” does and does not promise #

Conjugate action guarantees a constant velocity ratio, which is a strong statement and a narrower one than it sounds.

It says nothing about the torque ratio in a real mesh, because friction on the sliding flanks takes a cut whose size depends on where in the engagement the pair happens to be. It says nothing about load sharing, because whether one tooth pair or two carry the load alternates at the contact-ratio frequency and the teeth deflect under load, which changes when the handover happens. And it says nothing about noise, which comes mostly from that handover.

The practical consequence is that high-quality gears are deliberately not involute at the tips. Tip relief — a few micrometres of material removed near the tip — makes the geometry worse and the behaviour better, because it lets the incoming pair take up load gradually instead of striking at full stiffness.

That is a good illustration of the limit of what this site computes. Every gear figure here is exact involute geometry with assertions to prove it, and a real production gear departs from that geometry on purpose, by an amount chosen from the load. The geometry is the thing that must be right before the deliberate departures make any sense — the same relationship the whole site has to reality.

Generating an involute without knowing what it is #

The involute has a property that made it manufacturable a century before anyone could compute one, and it is the reason gear cutting works the way it does.

An involute gear tooth is the shape produced by a straight-sided rack rolling on the pitch circle. That is not an approximation or a coincidence — it is the geometric dual of the involute’s definition, and it means a gear cutter never has to know the involute’s equation.

The consequences run right through gear manufacturing.

One cutter per module and pressure angle. A hobbing tool is a rack in helical form, and the module is the number that makes gears interchangeable. The same hob cuts a twelve-tooth pinion and a two-hundred-tooth wheel, and the flank shapes it produces are different — correctly different, each the involute for its own base circle — without any change to the tool.

The pressure angle is a property of the tool. It is the flank angle of the rack, which is why 20° is a standard rather than a calculation, and why moving to 25° is a decision about a whole shop’s tooling, and why changing it means changing the tooling for a whole shop.

Profile shift is a machine setting. Moving the hob radially in or out generates a tooth from a different part of the involute, which changes the tooth thickness and the operating pressure angle without changing the tool at all. That is the whole of profile-shift correction, and it is available for free on any generating machine.

Undercutting is a generation phenomenon. The cutter tip sweeps below the base circle on a small gear and removes flank material the tooth needed. The seventeen-tooth rule is a statement about this process, not about the involute in the abstract.

So the involute won partly on its own merits and largely because the process that produces it is indifferent to tooth count, which no other conjugate curve offers.

What the assertions here actually cover #

It is worth being explicit about what the gear figures on this site prove and what they take on trust, since a curve that looks like a gear tooth is easy to draw and hard to check.

Proved by measurement: the flank points all lie outside the base circle; the normal to the drawn polyline is tangent to the base circle within a tolerance that a wrong flank fails; the base-radius ratio equals the tooth-count ratio; the contact ratio computed from the geometry is in range and consistent with the drawn path of contact; the undercut threshold classifies gears on either side of it correctly.

Taken on trust: nothing about loads, stresses, deflection, lubrication, thermal effects or manufacturing tolerance. The figures are exact geometry, and a real gear departs from exact geometry deliberately.

The tolerance on the tangency check deserves a note, because a tolerance chosen to make a test pass is not a test. It is bracketed from both sides: a correct involute sampled the way these figures sample it gives 2.0 × 10⁻⁵, and a flank approximated by a circular arc — the classic draughtsman’s substitute — gives 7.6 × 10⁻². The tolerance sits at 2 × 10⁻⁴, between them, so the check passes what is right and fails what is nearly right. A test that only passes correct input is half a test; proving it also rejects is the other half.

The two flanks use the same curve at different places #

The base circle sitting below the root has a consequence for the mesh that is easy to pass over, and it explains a fact about gear failure that is otherwise attributed to loads alone.

An involute’s curvature is not uniform. Close to its base circle the curve turns sharply; far from it the curve flattens, approaching a straight line. So where along the involute a tooth’s working flank sits decides how curved that flank is, and that position is set by the ratio of the tooth’s own radius to its base radius.

A large wheel has a large base circle, and its working flank sits at a radius only slightly larger — a stretch of involute far along the unwinding, where the curve is nearly straight. A small pinion has a small base circle and a working flank running from just outside it, which is the sharply-curved beginning of the same construction.

So the two members of a mesh carry the same kind of curve used at very different parts of it. Their flanks are not two versions of one shape; they are one shape sampled at two scales, and the pinion always gets the sharp end.

That is the geometric half of a fact every gear engineer knows: the pinion is the part that fails. A sharply curved flank against a nearly straight one concentrates the contact into a smaller patch, and the pinion also meshes more often — once per revolution of the wheel divided by the ratio. Two effects, both against the small member, and only the second is usually given.

It is also why profile shift is applied to the pinion rather than to the wheel. Moving the cutter out shifts the pinion’s working flank further from its base circle, onto a flatter stretch of the involute, which is precisely the geometry the wheel has for free. The remedy is not adding material; it is moving the tooth onto a better part of the curve it was already made of.

The law of gearing as a general statement #

The law is usually met in the context of gear teeth, and it is more general than that, which is worth stating because the generality is what makes it a law rather than a design rule.

Two bodies in contact transmit motion at a constant angular velocity ratio if and only if the common normal at the contact point passes through a fixed point on the line of centres. That says nothing about teeth, involutes, or gears. It is a statement about any pair of contacting profiles.

Reading it as a design instruction gives a construction rather than an answer: pick one profile freely, and the conjugate profile is determined. That is how non-circular gears are designed, how a cam and its follower are related, and how the flanks of a rotary pump are laid out. The involute is the special case in which both profiles are the same curve and the fixed point is where the pitch circles touch.

Reading it as a test is what this site does. Given a drawn flank, sample it, compute the normal at each sample, and check that the normal passes where it must — which for an involute means being tangent to the base circle. That is a measurement on the polyline that will actually be rendered, so it catches a flank that is nearly right as well as one that is wrong.

The general form also makes clear what a non-circular gear is doing. Elliptical gears, and the lobed pairs used in flow meters and quick-return drives, satisfy the law with a fixed point that moves along the line of centres, so the ratio varies through the revolution in a specified way. That is a gear pair with the varying ratio a linkage has, arranged deliberately, and it shows that constant ratio is a choice within the law rather than a consequence of it.

Which leaves the involute’s real distinction where this essay put it: not that it satisfies the law, but that it satisfies it while tolerating a change in centre distance — the one property no other conjugate pair offers, and the one that made gearing manufacturable.