Machine Kinematics
solved, then drawn
Teeth

Why a tooth is an involute

A gear tooth is not a shape anybody chose for its looks. It is what one requirement forces — that the ratio of the two shafts' speeds stays exactly constant while the contact point slides along the flank. Impose that and the curve is essentially determined.
16 min read 6 figures The constant that is not constantTwo routes to the same number

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.

The involute, unwoundHold a string taut against a circle and unwind it: the end traces this curve. Two properties follow immediately and between them they are the whole of gear geometry. The string is always tangent to the base circle, and it is always perpendicular to the curve. So the normal to an involute at any point is a tangent to its base circle — measured on the generated points here, to 2.3e-5 — and when two involutes touch, the common normal is a line tangent to both base circles, which does not move as the gears turn.the traced pointtangencybase circle r = 9.4020 teeth, module 1, 20° pressure anglenormal meets the base circle to 2.3e-5
Fig. 1 The curve that satisfies it. Unwind a taut string from a circle and its end traces an involute. Drag the slider to unwind further.

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 rbr_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 rbtr_b t. The first version did that and reported a 94% error on a perfectly correct flank.

20 teeth driving 32Both flanks generated from the involute, not approximated. The orange line is the line of action — tangent to both base circles, and the only place contact happens. Its length between the two tip circles divided by the base pitch is the contact ratio, 1.612 here, which means that for 61% of the cycle two tooth pairs are carrying the load and for the rest just one. The velocity ratio is 0.6250, and it is constant because the common normal never moves.pitch pointline of actionmodule 1, 20° pressure angle, centre distance 26contact ratio 1.612
Fig. 2 Two involutes meshing. The orange line is the common normal — tangent to both base circles — and the contact point travels along it. Drag to turn the gears and watch it stay put.

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 rb=rcos⁡αr_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.

Undercutting, either side of 17.10 teethFive gears, drawn whole and scaled to a common pitch circle so the tooth counts can be compared by counting them. The dedendum sits a fixed 1.25 modules below the pitch circle and the base circle sits at r·cos α, so as the tooth count falls the base circle rises relative to the root. Below N = 2/sin²α = 17.097 it rises above it, and the part of the flank between them lies where no involute exists — the cutter removes it. The familiar rule says seventeen; the exact figure is 17.10, so seventeen undercuts slightly and eighteen is the smallest count that does not. The teeth are not to a common scale, because at a common module the small gears would be unreadable.10 teethundercut14 teethundercut17 teethundercut18 teethclean24 teethcleanred: the root circle has risen above the base circlethreshold N = 2/sin²α = 17.097
Fig. 3 What happens when the root rises above the base circle: part of the flank is in a region where no involute exists, and the cutter removes it. The threshold is a formula rather than a preference.

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.

A ratio that is not a numberThe output-to-input angular velocity of a four-bar, computed from the velocity solution at every position. It runs from -0.398 to 0.333 — it changes sign, because the rocker turns back — with a mean of 0.000 that no instant of the cycle actually exhibits. Quoting a single figure for a linkage's ratio is quoting the average of that curve. A gear pair is the case where the same phrase is honest: its ratio is 0.5000 and stays there, which is not a coincidence but the property the involute was invented to guarantee.-0.25000.2500.5000100200300crank angle (degrees)output ÷ input angular velocitya 20:40 gear pair, 0.500mean 0.000four-bar 4/1/3.5/3-0.40 to 0.33 through one turn
Fig. 4 The contrast worth keeping. A gear pair’s ratio is genuinely a number and stays there. A four-bar’s varies through the turn and changes sign, and quoting one figure for it is quoting an average of that curve.
Ratios, and the gear that does not change oneThe ratio of a train is the product of its stages, so the middle gear of a simple three-gear train cancels: 30/20 × 40/30 = 40/20, exactly as if it were not there. What it does change is the direction, since every external mesh reverses — which is the entire reason idlers exist. A compound train, where two gears share a shaft, does not cancel, and that is how large reductions are built without absurdly large wheels.reduction ratio20 → 402.00 : 1reversed20 → 30 → 40 (idler)2.00 : 1same direction20 → 40, 15 → 45 compound6.00 : 1same direction20 → 60, 20 → 60, 20 → 6027.00 : 1reversedproduct of the stages · sign flips at every external meshthe idler cancels exactly
Fig. 5 What constant ratio buys once it exists: trains that multiply exactly, and idlers that cancel exactly. Neither would be true if the ratio wobbled through each mesh.
Sun 24, ring 72, planet 24An epicyclic train has three shafts and one equation relating them, so fixing any one gives a different ratio from the same gears. With the ring held, a sun input turns the carrier at 1 + 72/24 = 4.000 to one. Willis's equation and the tabular superposition method are independent derivations and both are computed here; the site requires them to agree, because planetary ratios are the most commonly mis-stated numbers in mechanism work and the sign errors are notorious.ringsunhold the ringsun in, carrier out4.000 : 1same directionhold the carriersun in, ring out−3.000 : 1output reverseshold the sunring in, carrier out1.333 : 1same directionWillis: (ω_s − ω_c)/(ω_r − ω_c) = −72/24both derivations agree, and the build requires it
Fig. 6 And the arrangement that gets three ratios from one gearset, which is only sensible because each individual mesh has a ratio that does not vary.

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 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.

Where this goes next

pitch pointline of actionmodule 1, 20° pressure angle, centre distance 26contact ratio 1.612 What happens in a mesh Contact between two gear teeth happens only along one straight line, and only over part of it. How much of that line lies between the two tip circles, divided by the base pitch, is the contact ratio — and if it drops below one the drive periodically stops being driven. 10 teethundercut14 teethundercut17 teethundercut18 teethclean24 teethcleanred: the root circle has risen above the base circlethreshold N = 2/sin²α = 17.097 Undercutting, and the seventeen-tooth rule Below a certain tooth count a standard cutter eats into the flank it is supposed to be forming. The count is quoted as seventeen. The formula gives 17.097, which means seventeen undercuts slightly and eighteen is the smallest that does not — the rounding went the convenient way. -0.25000.2500.5000100200300crank angle (degrees)output ÷ input angular velocitya 20:40 gear pair, 0.500mean 0.000four-bar 4/1/3.5/3-0.40 to 0.33 through one turn The ratio that is not a number A four-bar's output-to-input speed ratio runs from −0.29 to 0.51 through one turn and changes sign on the way. Quoting a single figure for it quotes the average of that curve, which the mechanism never exhibits. A gear pair is the case where the same phrase is honest, and it is honest by construction.