A tooth flank is an unwound strand

The involute is usually introduced as a curve: a parameterisation, a base circle, a roll angle, and the observation that a pair of them keeps a constant velocity ratio. The construction that gives it its name is mentioned and then dropped — involutus, rolled up, the curve a string traces as it is unwound.

That construction is not decoration. Read properly it says that the whole of involute gearing is a statement about a taut strand, and it explains the one property gears are chosen for in a way the curve’s algebra does not.

The curve, unwound #

Wrap a strand on a circle of radius $r_b$, hold the free end taut, and unwind. At roll angle $t$ the strand leaves the circle at the point $r_b(\cos t, \sin t)$, its taut portion is tangent there, and its free length is the arc it has given up: $r_b t$.

The free end is therefore at

$$r_b(\cos t + t\sin t,\; \sin t - t\cos t),$$

which is the involute — the same expression the gears field builds a flank from.

Measured against the gears field’s own involute across the whole flank, at three base radii:

The two routes are the same curve. That was never in doubt; what is worth having is that the strand’s version is built out of a tangent length and an arc length, which are this field’s two quantities, and it never mentions a curve at all.

The line of action is the strand #

Now the part that is not a restatement.

Put two base circles on fixed centres and cross a strand between them. The tangency solve puts that strand tangent to both, so its perpendicular distance from each axis is that axis’s own base radius — by construction, since that is where the tangency was placed.

A strand does not slip on what it wraps. So the two bodies must turn in the ratio of their arms, which are the two base radii, and nothing else can enter it.

That crossed tangent is exactly the line of action. Every contact between two involute flanks lies on it; the common normal at every contact is along it; the pressure angle is its inclination. All of which the gears field derives from the involute’s geometry, and all of which the strand has by construction.

The check is that the arms are the base radii, to $1 \times 10^{-10}$ mm at every centre distance tested — measured as the perpendicular distance from each centre to the drawn segment rather than taken from the circle that produced it.

Why the shafts may move #

Here is the property involute gearing exists for, and the reason to have gone through the strand.

Push the two axes apart. The strand’s direction changes; its length changes; both tangency points move round their circles; the pressure angle rises. The arms do not move, because the arms are the base radii and the base circles have not changed.

The ratio’s spread across those is exactly zero — not a small number, the same double-precision value three times.

That is the belt’s own indifference. A belt on two pulleys has a ratio set by the two radii and does not care where the shafts are, which nobody finds remarkable; the involute inherits it because the involute is the curve whose contact is a crossed strand on two circles that never change size.

So the usual account has the dependency backwards. The involute is not a curve that happens to have a useful invariance; it is the curve that makes a gear pair behave like a crossed belt, and the invariance is the belt’s.

The number the curvature field already measured #

The strand between the two base circles is 41.0424 mm long at the nominal centre distance, and that number has appeared on this site before.

Two flanks, one law measures the two contacting flanks’ radii of curvature at every position of the mesh and finds that they vary fourfold each while their sum stays at 41.042 mm. That sum is this strand.

The reason is immediate once the strand is in view. A point on the line of action is at distance $\rho_1$ from one tangency and $\rho_2$ from the other, and those distances are the two flanks’ curvature radii, because an involute’s centre of curvature is its own base-circle tangency. The two add to the length of the segment between the tangencies, which is a fixed length. A quantity that had to be measured over a sweep is a segment’s length.

Internal gearing is the open belt #

The senses list settles a second thing for free.

An external pair is a crossed strand: opposite senses, the tangent passing between the circles, the two wheels turning opposite ways. The pressure angle satisfies $\cos\alpha = (r_{b1} + r_{b2})/a$ — the crossed tangent’s own relation, with the sum of the radii in it.

An internal pair is an open strand: the same sense at both, the tangent outside, the two wheels turning the same way. On a 24-tooth pinion in a 72-tooth ring at module 4, the arms come out at 45.105246 and 135.315737 mm — the two base radii — with a ratio of 0.333333 and an operating pressure angle of exactly 20.0000°, and the tangent between the two base circles is 32.834 mm rather than the external case’s 41.042.

That the internal pair’s line of action is shorter is the geometric reason internal gearing has a higher contact ratio than external gearing at the same tooth counts: there is less line, and the teeth are in contact over more of it. The strand gives the length and the tooth geometry does the rest.

The pitch point, and why it is where it is #

One more quantity falls out of the strand without being computed, and it is the one the law of gearing is stated in terms of.

The pitch point is where the two bodies’ material points have the same velocity — the one place on the line of centres where the mesh is rolling rather than sliding. For a crossed strand it is where the strand crosses the line of centres, and that is immediate: the strand’s speed is the same on both sides of the crossing, and the two arms there are the distances from each axis, so the two surface speeds match exactly where the strand’s own point does.

Its position divides the centre distance in the ratio of the base radii, which is the ratio of the pitch radii, which is the tooth ratio. All three of those are the same statement about the same crossed tangent.

That is worth noticing because the meshing field arrives at the pitch point by a different route entirely — solving for where the relative velocity of two touching bodies vanishes, and finding that the common normal at every solved contact passes through it to 2.2 × 10⁻¹⁴ mm. Here it is a crossing point of a line and a segment, with no solve of any kind.

What generation looks like from the strand’s side #

The rack that cuts an involute is the third appearance of the same tangent, and it closes the argument.

A straight-sided cutter rolling on the pitch line generates the involute — measured at 1.4 × 10⁻¹⁴ mm on tooth counts from 12 to 60, with nothing in the generation knowing what an involute is. In strand terms the cutter’s straight flank is the line of action: it is the tangent to the base circle, held at the pressure angle, and the wheel rolls under it.

So the same segment is three things at once, depending on what is put at its ends. Between two base circles it is the line of action of a mesh. Between one base circle and a straight edge it is the generating flank of a rack. Wrapped on one circle and let go, it is the involute itself.

That triple identity is the practical reason involute gearing displaced everything else. One straight-sided tool cuts every tooth count, because the tool is the line of action and the line of action is a tangent to a circle that the tooth count merely sizes.

What the strand does not give #

The correspondence is exact and it is a correspondence about one contact, which is a long way short of a gear.

A gear has teeth of finite thickness, so the flank exists only over part of the line of action, and how much of it is in contact is the contact ratio — a quantity about the addendum circles, not about the strand.

A gear’s flank stops at the base circle, and below it the tooth is a fillet with a trochoidal shape that no strand traces. Undercutting is the cutter removing part of the flank, which is a statement about a generating motion rather than about an unwinding.

And a gear pair slides. The strand’s contact rolls without slipping, and two involute flanks emphatically do not: they slide everywhere except at the pitch point, by up to 12 mm per radian at the ends of the contact. The strand model reproduces the normal geometry of the mesh exactly and says nothing about the tangential motion, which is where the wear is.

And why the cycloid is not a strand #

The comparison completes itself in the other tooth form.

A cycloidal pair’s contact path is not a line. Fitted rather than assumed, it is a circle of radius 24.000000 mm against a describing circle of 24, and a straight line misses those contacts by 0.11 mm.

A taut strand between two fixed circles is a straight segment. So there is no pair of bodies for which a cycloidal mesh’s contact path is a strand, and the analogy is not merely inconvenient — it is unavailable. Which is the same fact as the cycloid’s sensitivity to centre distance: its contacts do not lie on something whose arms are fixed, so moving the axes changes the ratio, by 2.9 × 10⁻³ mm of shape error at two hundredths of a millimetre where the involute answers with its own floor of 3 × 10⁻⁸.

What was known, and when #

The string construction is old and was understood as a construction rather than as an analogy. Euler wrote about the involute as a tooth form in the 1760s; the unwinding-string definition is older still, and it is why the curve is named for being rolled up.

What is worth restating is the direction of the argument. The involute was not deduced from the law of gearing — conjugate action rules almost nothing out, and a flank invented to be nothing in particular has a perfectly good mate. It was selected, from among all the shapes that would work, on properties the law of gearing does not mention: it is easy to generate with a straight-sided cutter, it tolerates centre-distance error, and its line of action is straight — and a rack generates it.

Every one of those three is a strand property. A straight line of action is a taut segment; a fixed ratio under a moving centre distance is a belt’s ratio; and generation by a straight rack is the same tangency condition seen from the tool’s side.

What the analogy is worth #

It would be easy to read all of this as a mnemonic. It is worth being specific about which parts of it are load-bearing.

It gives the right answer for the right reason. The centre-distance invariance is usually justified by an algebraic argument about the involute’s parameterisation, which is correct and explains nothing about why this curve. The strand explanation says the ratio is a quotient of two arms and the arms are fixed, and that argument would work for any pair of shapes whose contact normal is tangent to two fixed circles — which is a characterisation of the involute rather than a property of it.

It predicts the exceptions. The cycloidal pair’s contact path is a circle, so no strand lies on it, and the cycloid’s centre-distance sensitivity follows. That is a prediction rather than a re-description: a shape whose contacts are not a straight segment cannot have fixed arms, so it cannot be indifferent.

It connects the number to a length, which is the sum two flanks’ curvatures make. The two flanks’ radii of curvature summing to a constant is a measured invariant in one field and a segment’s length in this one.

What it does not do is replace the tooth geometry, and the boundary is clean: the strand model owns the normal direction of the mesh — where contact is, what the ratio is, where the pitch point is, how the pressure angle moves — and owns nothing about the tangential direction, the tooth’s extent, or what happens below the base circle.

Profile shift does not touch the strand #

The strand reading settles one more property of involute gearing immediately, and it is the property that makes the tooth form practically flexible rather than merely mathematically tidy.

Profile shift is the standard remedy for a small pinion: move the cutter outward when generating, and the tooth comes out thicker at the root and thinner at the tip, with undercutting avoided at counts below the seventeen the gears field measures. It changes the tooth’s shape and its thickness and where along the involute the flank sits.

What it does not change is the base circle. The base radius is set by the module, the tooth count and the cutter’s own pressure angle, and the cutter’s radial position appears nowhere in it. So the strand is the same strand, wrapped on the same two circles, and the crossed tangent between them is the same line.

From which everything follows in one step. The line of action is unchanged, so the ratio is unchanged; the arms are still the base radii, so the ratio is still the ratio of tooth counts; and a shifted pair meshes with an unshifted one of the same module and count at exactly the same ratio. That is a statement people usually reach through a paragraph about how the operating pressure angle changes to compensate, and the strand gives it without any compensation at all — nothing changed that the ratio depends on.

The same argument covers the centre distance, which the essay has already measured at exactly zero spread, and it covers them together: a shifted pair run at a modified centre distance still has both base circles where they were, so still has the same strand and the same ratio. The two remedies compose freely for the same reason neither of them does anything on its own.

What does change is everything the strand cannot see. The length of the contact path changes, because the flanks now occupy different stretches of the involute, so the contact ratio changes and a shift taken too far leaves fewer than one tooth pair in contact at some instants. The tooth thickness changes at every radius. The tip and root clearances change. All of those are consequences of where the flank starts and stops along the line of action, which is a question about tooth boundaries rather than about the curve.

That division is the clean statement of what the strand model is good for, and it is the same division the essay’s last section draws. The strand decides the ratio and the geometry of the contact path; the tooth’s boundaries decide everything else. Profile shift is the clearest case, because it moves the second and leaves the first exactly where it was.

What is not modelled #

Nothing here is a tooth. There is no thickness, no addendum, no fillet, no backlash and no profile shift, and every one of those is a decision the strand model cannot express — it has one contact and a gear tooth has two flanks and a root. The strand is inextensible and the flanks are rigid, so nothing in this essay says anything about load sharing between the one, two or three pairs of teeth that are in contact at a given instant. And the pressure angle here is a geometric inclination and not a force direction: what fraction of a transmitted force acts along the line of action is a statics question, and the whole of this correspondence survives with every force unknown.