The tool is the definition

A gear tooth is an involute, and an involute is a transcendental curve: a taut string unwound from a circle, with no centre and no radius, described by a parameter that appears both inside and outside the trigonometric functions. It is exactly the sort of shape that is awkward to make.

The cutter that makes it has no curve on it at all. It is a straight edge at twenty degrees to the vertical, and the involute appears because of the way the edge and the blank are moved, not because of anything the edge is shaped like.

One number describes the whole process #

The rack’s pitch line rolls on the wheel’s pitch circle without slipping. That single condition is the entire kinematic content of gear cutting: the rack advances $r$ millimetres for every radian the blank turns, where $r$ is the pitch radius. Nothing else enters — not the tooth count, not the module, not the pressure angle. Those are all consequences of the rack’s own shape and of that one rolling condition.

Feed the pair to the meshing equation — the rack’s straight flank as the given profile, the sliding rack and the turning blank as the two motions — and at each position it returns the point of the flank in contact. Collect those contacts in the blank’s own frame and the result is the tooth.

The claim to be checked is that this curve is the involute of the circle $r\cos\alpha$. The check that does not smuggle in the answer is the involute’s own defining property: the normal at any point of an involute is tangent to its base circle, so the perpendicular distance from the wheel’s centre to the normal at every generated point should equal $r\cos\alpha$ exactly.

Measured over 240 contacts on a 24-tooth wheel of module 4: worst departure $1.4\times10^{-14}$ mm, on a wheel 48 mm in pitch radius. Over five tooth counts from 12 to 60, the worst anywhere is of the same size. The generation was handed a straight line, a rolling condition and a dot product. It gives back the involute.

What the rack actually is #

It is worth writing the cutter down, because every number in the standard for a gear is a dimension of it rather than of any wheel.

The rack has a pitch line; teeth of thickness half the pitch stand out from it towards the blank, and gaps of the other half take the wheel’s teeth. Each flank is a straight line inclined so that its normal makes the pressure angle $\alpha$ with the direction of the rack’s travel. The tooth stands $h_a$ proud of the pitch line and the gap runs $h_f$ behind it — one module and one and a quarter modules in the usual proportion, so that a wheel’s tip clears the mating wheel’s root.

In the rack’s own frame, with $s$ measured from the pitch line towards the blank, one flank is

$$p(s) = (x m - s,\; s\tan\alpha),$$

where $x$ is the profile shift and is zero for a standard wheel. There is no other curve in the file: the whole cutter is that line, its mirror image, a tip that joins them and a root that joins the next tooth.

Twenty degrees is a compromise and not a theorem. A larger pressure angle makes the tooth stubbier and stronger at the root, allows fewer teeth before the cutter starts eating the flank, and pushes harder on the bearings; a smaller one runs more quietly and needs more teeth. Fourteen and a half degrees was the older standard, twenty-five is used where strength matters more than noise, and the essay on where the corner overtakes the involute measures what each of them costs in tooth count.

The contacts fall on a line, and the line finds the base circle #

There is a second thing the generation produces that nobody asked it for, and it is the more striking of the two because it is a property of the process rather than of the tooth.

Every contact is a point of the fixed frame. Plot them all — not in the wheel’s frame, where they trace the flank, but in the frame the machine sits in — and they lie on a straight line.

Fitted by total least squares to the 240 contacts, the worst departure from the line is $1.1\times10^{-14}$ mm. That line is the line of action, and the classical account introduces it as a construction — draw the common tangent to the two base circles — rather than as a measurement.

Drop a perpendicular from the wheel’s centre onto the fitted line. Its foot lands at $45.105246$ mm from the centre; $r\cos\alpha$ for this wheel is $45.105246$ mm. The two agree to the ninth figure, and the second number was never used in producing the first.

That is where the base circle comes from. It is not a construction line somebody chose in order to define the involute; it is the circle the contacts of a straight edge are tangent to, and every other property of the tooth follows from it. The involute is the curve that has that circle as its evolute, and the reason a gear tooth has one is that a straight edge dragged past a turning blank produces one.

The fillet is a different kind of curve #

The rack’s flank is a straight segment and it ends: at the top in the tip corner, at the bottom in the root. When the contact reaches the corner, the generation of the flank stops, because there is no more flank to be in contact with.

What happens below that point is that the corner goes on cutting. A corner is a single point, and the envelope of a family of points is nothing more than the family itself — the corner sweeps out its own trajectory in the wheel’s frame, and that trajectory is the fillet at the root of the tooth.

The fillet is therefore a trochoid — the path of a point rigidly attached to a body whose pitch line rolls on a circle — and not, as gear drawings often show, a circular arc. Getting that right matters for two reasons. It is where a tooth breaks, so its actual curvature is a stress-raiser somebody has to know; and, on a wheel with few teeth, the trochoid crosses into flank the straight edge has already generated, which is undercutting and is the subject of its own essay.

For the same reason, an involute drawn continuing below the base circle is a drawing error rather than a simplification. There is no involute below the base circle — the string has run out — and what is actually there is a piece of trochoid whose shape depends on the cutter, not on the wheel.

The flank is an arc, and both its ends are the tool’s doing #

The generated flank does not run from the centre of the wheel to infinity. It starts and stops, and both limits are properties of the cutter rather than of the involute.

At the top, the flank ends where the blank ends: the tip circle is turned before the teeth are cut, and the involute simply stops there. That end is a diameter somebody chose.

At the bottom, the flank ends where the contact runs off the rack’s straight edge and onto its tip corner. Past that instant there is no straight flank in contact, so no more involute is generated, and the corner’s trochoid takes over. On a wheel with plenty of teeth that transition happens at or below the base circle, and the whole usable involute survives; on a wheel with few teeth it happens above the base circle, and the corner then comes back through flank it has already made.

So the usable part of a tooth is bounded above by a turning operation and below by a property of the cut, and the arc between them is what has to be long enough for the next tooth pair to pick up before this one lets go. That length, divided by the base pitch, is the contact ratio, and it is the number that decides whether a gear train is continuous or is a series of impacts.

Why generate rather than form-cut #

A tooth space could be cut with a shaped tool: grind a cutter to the exact shape of the gap, plunge it in, index the blank, repeat. That is called form cutting, it is perfectly possible, and it is what a hobby workshop does with a set of eight involute cutters.

Generating is better for reasons that are all consequences of the tool being straight.

One tool per module. The rack’s shape depends on the module and the pressure angle and on nothing else; the tooth count enters only through the rolling radius. So one straight-sided cutter makes every wheel of that module, and any two wheels it makes will mesh with each other. A set of form cutters is a compromise — the usual eight cover the whole range of tooth counts by being approximately right for a band of them.

The accuracy of a straight line. The tool has to be made to a shape, inspected against that shape, and re-sharpened without losing it. A flat and an angle can be ground and measured to a precision a curve cannot, and re-sharpening a straight-sided tool on its front face leaves the generating geometry untouched.

Corrections are motions rather than shapes. A profile shift is the same tool held further out; a helical gear is the same tool fed at an angle; a tapered tooth is the same tool with a varying offset. Every one of them changes the motion while leaving the cutter alone, which is why gear standards are written as descriptions of a rack rather than as families of curves.

What the generation costs to compute #

A note on the arithmetic, because it is the reason this field can afford to draw what it draws.

Each contact is one root of one scalar equation, found by sampling the residual along the rack’s flank, bracketing every sign change and closing each bracket by bisection. That is about six hundred evaluations of a dot product per position — trivial — and it has two properties worth the cost over a Newton solve. It cannot miss a root the sampling saw, which matters because a profile can carry more than one contact at a time and a solver that returns the first root would silently drop the others. And it cannot leave the bracket it was given, so it cannot jump to a neighbouring branch and return a curve stitched together out of two different contacts, which is a failure that produces a smooth and entirely wrong shape.

The whole tooth is then a loop over positions, and a wheel is a loop over the tooth. There is no optimisation anywhere in it, no fitting, and no curve library: the flank in the figures above is the list of roots, drawn.

What a real hob does that this does not #

The generation here is planar and kinematically exact, and a real gear-cutting machine differs from it in ways worth naming so the model is not read as a description of a workshop.

A hob is a worm with cutting edges rather than a rack, so the straight edges are arranged along a helix and the “rolling” is a coupled rotation of hob and blank rather than a slide. The tooth it generates is the same one — a hob is a rack wrapped round a cylinder — but the machine’s business is the coupling ratio between two rotations rather than a rack’s linear feed.

A shaper uses a cutter that is itself a gear, and the same envelope argument applies with a different given profile: the generated shape is conjugate to the cutter, and the cutter is conjugate to a rack, so everything meshes. It can cut internal gears and up to a shoulder, which a hob cannot.

Neither machine cuts continuously in the way the model does: the tool has a stroke, the cut is interrupted, and the finished flank is a series of tangent facets whose size is set by the feed. The facets are the difference between a geometric flank and a manufactured one, and their height is a machining parameter rather than a property of the geometry.

And none of that is force. What the cutting takes, how the tool wears, what the surface finish is: outside, as everywhere on this site.

Two descriptions of the same tooth #

There is a small confusion this construction disposes of neatly. A gear tooth can be described in two vocabularies, and beginners are often taught both without being told they are the same object.

The base-circle description says: the flank is the involute of a circle of radius $r\cos\alpha$, the line of action is tangent to that circle at the angle $\alpha$, and the ratio is the ratio of the base radii. Everything in it is a property of the wheel.

The rack description says: the tooth is what a straight edge at $\alpha$ leaves when its pitch line rolls on a circle of radius $r$, with a tooth thickness of half the pitch at that circle. Everything in it is a property of the cutter and of a rolling.

They describe the same curve, and the measurement above is the bridge: generate from the rack, measure the base circle off the contacts, and the two agree to nine figures. Which vocabulary is the useful one depends on what is being decided. A meshing question — where the contact is, what the ratio is, what happens when the centres move — is easiest in base-circle terms. A manufacturing question — what cutter, what shift, what tooth thickness, what happens if the tool is set wrong — is easiest in rack terms, and the standards are written that way for exactly that reason.

The idea underneath #

The generating principle is easy to state once the machinery is in place, and it is worth stating in its general form because it is what makes the rest of this field work.

A shape that is hard to describe can be produced by a shape that is easy to describe, if the motion between them is chosen well. The involute is hard; a straight edge is easy; the rolling of a line on a circle is easy; and the involute is what the three of them make between them.

That is the same trade the whole subject of mechanism runs on, from the other end. A straight-line linkage makes a straight motion out of pin joints because a straight guide was hard to make and a pin was not. Here the straightness is in the tool and the awkward curve is in the product. Both are the same manoeuvre: put the difficulty where the workshop can handle it, and let the geometry move it somewhere else.

The rung’s title deserves one more turn, because the tool is the definition is a stronger statement than it first reads. A gear is not a shape that a tool approximates; the shape is defined by the tool and the motion, and the involute is a description of the result rather than a specification of it. That inverts the usual relationship between a drawing and a part. Everywhere else on this site a mechanism has a nominal geometry and manufacture approaches it; here the nominal geometry is what the process produces, so the flank has no error in the ordinary sense — a wrongly cut tooth is a tooth cut by a rack in the wrong position, which is a different exact involute. That is why gear inspection reports profile error against a nominal involute rather than against a drawing, why one cutter makes every tooth count, and why profile shift is a machine setting rather than a redesign. A part whose definition is a process has its tolerances on the process, and the involute’s whole industrial history is a consequence of being that kind of part.