A tooth that lives on a sphere

Two shafts that are not parallel cannot be joined by two cylinders. They can be joined by two cones with a common apex, and the whole of bevel gearing follows from that one substitution — including the single place in this field where the tooth a machine is actually cut to is not the tooth the geometry asks for.

The cones are forced, and the angles are arithmetic #

Take two shafts meeting at an angle Σ. For a pair of bodies to roll on one another without slipping, the surfaces in contact must have zero relative velocity along the contact — which for rotation about two intersecting axes means the contact must lie on the line through the intersection about which the relative motion is a pure rotation. That line is fixed in space, and the surfaces swept by it about each axis are cones.

So the pitch surfaces are cones, their apexes coincide with the shaft intersection, and their half-angles satisfy

$$\delta_1 + \delta_2 = \Sigma, \qquad \frac{\sin\delta_2}{\sin\delta_1} = \frac{z_2}{z_1}.$$

For a twenty-tooth pinion driving forty at a right angle that gives 26.565° and 63.435°, and the sines are in the ratio 2.0000 exactly. The second relation is the cone version of the pitch-circle relation: two cones roll when their pitch radii at any common distance from the apex are in the ratio the counts demand, and the sine is how a cone’s radius depends on its angle.

What the two relations do not do is say anything at all about the teeth. Exactly as in the flat case, the pitch surfaces carry the ratio and the teeth are what makes a pair of them turn each other.

The tooth belongs on a sphere, and that is the difficulty #

Every point of a bevel tooth is at a fixed distance from the apex, because both cones share that apex and the tooth is cut into both of their surfaces. So the tooth’s profile — its cross-section at a given distance from the apex — lies on a sphere centred there.

Everything in the planar theory has a spherical counterpart. The pitch circles become circles on the sphere, the base circles become circles on the sphere, and the involute becomes the spherical involute, generated by unwrapping a great-circle arc from a small circle instead of unwinding a straight string from a circle. The law of gearing holds in the same form. Nothing about the argument fails.

The difficulty is not geometry, it is manufacture. A sphere is not developable. No piece of it flattens into a plane without stretching, which is a theorem about curvature and not a limitation of any process — and it means no flat-sided cutter, no rack, no straight generating edge produces a spherical involute exactly.

That is a large statement and it deserves its consequence spelled out. One rack cuts every flat wheel, and that single fact is what makes a gear standard possible: a shop owns one cutter and every gear it makes will run with every other. There is no such rack for bevels. The tool has to be a different shape for every cone angle, and what a bevel generator actually does is approximate.

Tredgold’s substitution, and what it costs #

The approximation everybody uses is two hundred years old and is one line long: replace the sphere by the cone tangent to it along the pitch circle.

That cone is the back cone — its elements are perpendicular to the pitch cone’s, so it meets the sphere at right angles along exactly the circle the teeth are spaced round. A cone is developable. Rolled flat it becomes a sector of a plane, the teeth in it become ordinary planar involutes, and the whole planar theory applies unchanged.

The back cone’s slant length from its own axis to the pitch circle is $r/\cos\delta$, so the equivalent spur gear has pitch radius $r/\cos\delta$ and therefore

$$z_v = \frac{z}{\cos\delta}$$

teeth. For the twenty-tooth pinion at 26.565° that is 22.361, and for its forty-tooth wheel at 63.435° it is 89.443. Neither is an integer and neither has to be: no gear is cut to $z_v$. It is a shape parameter, and what it decides is the curvature of the flank.

How wrong the substitution is, in modules #

The site’s habit is that an approximation gets a number rather than an adjective.

The back cone touches the sphere along the pitch circle and leaves it above and below. A point a distance $t$ up the back-cone element from that circle sits at $\sqrt{R^2 + t^2}$ from the apex where the sphere is at $R$, so the departure is

$$\sqrt{R^2+t^2} - R \;\approx\; \frac{t^2}{2R},$$

quadratic in how far up the tooth the point is and inversely proportional to the cone distance.

At the tip of a tooth — one addendum, so $t = m$ — a bevel with a cone distance of ten modules departs by 0.0500 modules and one of thirty modules by 0.0167. The quantity has no pressure angle in it, no tooth count and no units: it is a pure ratio of the tooth to the cone.

That is why the construction survived. A bevel of ordinary proportions — twenty-odd modules of cone distance — carries a profile error a fiftieth of a module, which is smaller than the tolerance the teeth are cut to and very much smaller than the backlash they are assembled with. A squat bevel with a cone distance of six modules carries 0.083, which is not.

It also explains the standard’s face-width rule. The face is conventionally limited to about a third of the cone distance, and the usual justification is that the small end’s teeth get too small to cut. The error argument gives a second reason from the same ratio: the whole tooth geometry is fixed at the large end, and the further inward the face runs, the further the real geometry is from the one the construction was done at.

The small end is a different gear #

One more consequence of the cone deserves its own paragraph, because it has no counterpart at all in the flat field.

A spur gear’s tooth is the same all the way across its face. A bevel gear’s is not: every dimension shrinks in proportion to the distance from the apex, so the module at the inner end of a face of width $b$ is $(R-b)/R$ times the module at the outer end. At the conventional face width of one third of the cone distance, the inner module is two thirds of the outer one — the tooth at the small end is a third smaller in every dimension than the tooth at the large end, on the same gear.

That is why bevel proportions are always quoted at the large end, and why the face-width limit is a limit rather than a preference. Run the face to half the cone distance and the inner module is half the outer; run it to the apex and the teeth vanish. The rule of thumb is that a bevel’s face may be a third of its cone distance, and the arithmetic behind it is this ratio rather than anything about stress.

It also compounds with the error above in the direction a designer would hope it did not. The whole tooth geometry is set at the large end, where the departure is smallest relative to the tooth; the small end has the same absolute cone distance but a smaller tooth, so its tooth is a smaller fraction of the sphere and the approximation is better there. The two effects go opposite ways, which is why the face-width rule is about the teeth getting small rather than about them getting wrong.

What a bevel generator is actually shaped like #

There is a second half to the manufacturing story, and it explains why the crown wheel turned up twice in this essay before it had been introduced.

A spur gear is generated by rolling a rack against the blank, and the rack is the member of the spur family whose pitch surface is flat. A bevel gear is generated the same way, by rolling against the member of the bevel family whose pitch surface is flat — which is the crown wheel. In practice the tool is a cutter head carrying blades that sweep out one tooth of an imaginary crown wheel while the blank rolls against it, so a bevel generator is a machine for pretending to be a crown gear.

That is the same logic the rack argument runs on, one field over: a standard specifies the flat member of the family and defines every other member as whatever meshes with it. The difference is that the flat member of the spur family has straight flanks and generates the exact curve, and the flat member of the bevel family has straight flanks and does not.

What a straight-bladed crown cutter produces is not the spherical involute. It is a different curve — one whose line of contact, traced on the sphere, makes a figure-of-eight, and which is named for that shape. It is conjugate to itself, so two gears cut by the same head do mesh correctly with one another, which is the property a standard actually needs; it is simply not the curve the spherical law of gearing singles out. Drawn here is the back-cone approximation, and that curve is not computed, and the distinction is named here rather than absorbed into the word “approximate”, because the two approximations have different sources: one is a cone standing in for a sphere, the other is a straight blade standing in for an arc.

Both are small and neither is zero. What can be said firmly is the direction: every step from the sphere toward something a machine can hold is a step away from exactness, and the flat case has no such steps in it at all.

The equivalent count is the number that matters #

Once $z_v$ is in hand, everything in the flat theory carries over with $z$ replaced by $z_v$, and two consequences are worth having.

A bevel pinion may be smaller than a flat one. Undercutting is a condition on the equivalent count, so a real bevel gear is clean from $z \ge (2/\sin^2\alpha)\cos\delta$ teeth upward. At the 26.565° cone above that is 15.29 teeth against the flat gear’s 17.10; at the 45° cone of a one-to-one pair it is 12.09; and it keeps falling as the cone steepens. The familiar seventeen is a fact about a flat gear and about nothing else.

A bevel pair carries more contact than the spur pair of the same counts. The contact ratio is computed on the equivalent gears, which are larger, so it comes out higher: 1.712 for the 20–40 pair above against 1.635 for a flat 20–40 pair at the same module and pressure angle. Both consequences run the same way, and both are the same statement — a bevel tooth is shaped like a tooth of a bigger gear than the one it is on.

What the error costs kinematically #

A profile error of a fiftieth of a module is a length. What a reader wants to know is what it does to the motion, because that is the quantity the field exists to protect.

The argument runs the way it runs for a non-involute flank. The law of gearing says the common normal at the contact must pass through the pitch point; a profile that departs from the conjugate one has a normal that misses, and the miss shows up as the driven shaft running fractionally fast and then fractionally slow within each tooth. Over a whole revolution the ratio is still exactly $z_2/z_1$, because that is a count — what the counting buys is immune to the shape. Within a tooth it is not.

So a bevel pair’s error is a periodic error at tooth frequency rather than a drift, which is the same category as chordal action on a chain and unlike a wrong tooth count, which is a permanent factor. That matters for what can be done about it: a periodic error at a known frequency is something a designer can budget for, and a mean ratio that is exact is something a designer can rely on.

The profile departure is not turned into an angular error here, because doing it honestly needs the conjugate curve computed on the sphere and compared against the one the back cone gives — which is exactly the calculation named and not done above. What it can say is which of the two kinds of error this is, and that the distinction is the one that decides whether it accumulates.

The crown wheel, where the equivalent gear is a rack #

There is a degenerate case, and unusually for a degeneracy it is a thing people build.

Let the cone angle go to 90°. The pitch cone becomes a flat disc, the back cone becomes a cylinder, and $z/\cos\delta$ runs away without bound. A spur gear of unboundedly many teeth is a rack — its pitch circle is a straight line, its base circle is a straight line, and its involute is a straight line at the pressure angle. So a crown wheel’s teeth are straight-sided, and that is not an approximation or a manufacturing convenience: it is what the limit says.

The crown wheel is the bevel field’s rack, in the same sense and for the same reason the rack is the spur field’s: the member of the family whose pitch surface is flat, whose teeth are straight-sided, and against which every other member is defined. It is also the one bevel whose teeth can be cut with a straight tool, which is why crown-and-pinion arrangements turn up in cheap mechanisms where a proper bevel pair would not be worth the tooling.

What was measured #

Three quantities, and each is computed from the cone rather than quoted.

The cone angles come from solving $\sin(\Sigma - \delta_1) = (z_2/z_1)\sin\delta_1$, and the figure prints the ratio of the sines beside the tooth-count ratio so the two can be compared: 2.0000 against 40/20 on the default pair. A pair of angles that satisfied the sum and not the sine ratio would be two cones that meet and do not roll.

The equivalent counts are $z/\cos\delta$, and the check on them is a direction rather than a value: $z_v$ must exceed $z$ at every cone angle, because a cosine is at most one. That is the requirement a cosine applied the wrong way up fails, and it fails at every angle rather than at an awkward one.

The departure is $\sqrt{R^2 + t^2} - R$, computed exactly and compared against the $t^2/2R$ the series gives. On a cone twenty-two modules long the two agree to five parts in ten thousand, which is a statement about the series and not about the gear — and it is the reason the approximate form can be quoted in an argument about how big the error is.

The claim the sweep supports is that the departure falls as the cone lengthens, checked at two cone distances a factor of four apart rather than at one. A quantity that did not fall would mean the tooth and the cone are not in the ratio the algebra says, which is the mistake a stray module would produce.

What is exact here and what is not #

It is worth separating the layers, because a field whose teeth are approximate still has exact things in it and the difference gets lost.

Exact: the ratio. $z_2/z_1$, by counting, as everywhere in this field. The cone angles do not enter it.

Exact: the cone angles. $\delta_1 + \delta_2 = \Sigma$ and $\sin\delta_2/\sin\delta_1 = z_2/z_1$ are geometry, not approximation, and they determine the two angles uniquely for any shaft angle.

Exact: the spherical involute. It satisfies the law of gearing on the sphere in the same way the planar involute does in the plane. If a bevel pair could be cut to it, its ratio would be constant within each tooth as well as over a whole turn.

Approximate: the profile actually cut, by $\sqrt{R^2+t^2}-R$ at each height up the tooth. That is the whole of it, and it is a manufacturing approximation rather than a kinematic one — the mathematics knows the right answer, and the tooling cannot reach it.

The distinction matters because the usual shorthand is that bevel gears “are not as accurate” as spur gears, which suggests a vagueness rather than a number. The number is small, it is computable before anything is cut, and it is a function of one ratio.

Where this connects to the rest of the collection #

Bevel geometry is the gears field’s meeting with the spatial one, and the meeting is closer than it looks.

A spherical four-bar is a linkage whose joints all pass through one point, whose bars are arcs and whose lengths are angles. A bevel pair is two bodies rotating about two axes through one point — the same setting exactly. Everything the spatial field says about motions on a sphere applies here, including the observation that every planar result carries over with a sine where a length used to be, which is precisely the form the ratio relation took above.

The other connection is the scale argument. A spur pair has one length in it and everything else is either proportional to that length or independent of it. A bevel pair has the same property, and the departure measured here is the exception that proves it: the error in modules depends on the ratio of tooth to cone and not on the size of either, so scaling a bevel gear changes nothing about how well Tredgold’s construction fits it. Making it stubbier does.

Still open: the count the cutter sees, again #

The equivalent count $z/\cos\delta$ is the second time in this field that a cutter has worked to a tooth count other than the one on the drawing, and the two occurrences have the same shape without having the same cause.

Here the cause is a cone: the tooth is spaced round a circle of radius $r$ and shaped to a circle of radius $r/\cos\delta$, and the cosine is the cone’s. The next essay meets the same substitution with a different cosine in it, arising from a tooth that is cut on a slant rather than on a cone — and there the exponent is a cube rather than a first power, which is a difference worth explaining rather than noting.

The other thing left open is an internal bevel, which exists, is rare, and would need the interference condition worked out on the equivalent counts rather than the real ones. Nothing here forbids it; nothing here computes it either.