Involute Gear Generator

Two smooth wheels pressed rim to rim turn each other at a perfectly steady ratio, right up to the moment they slip. Teeth stop the slip. The trouble is that teeth, shaped carelessly, also ruin the steadiness.

Picture square pegs on two wheels. Each peg would shove its partner forward in a lurch, then let go with a knock as the next one arrived. What a gear pair needs is a tooth shape that keeps the driven wheel turning at a constant fraction of the driver’s speed at every instant of every engagement, not merely on average over a turn. Profiles that manage this are called conjugate, and the condition is geometric. The line drawn square to the two touching flanks, the line of action, must always cross the line of centers at one fixed spot, the pitch point.

Many curves can satisfy that rule. Cycloidal teeth do, and clockmakers still use them. Nearly every other gear made today uses one curve instead, the involute, and the reason is less about elegance than about forgiveness.

The curve at the end of a string

Wrap a string around a tin can, tie a pencil to the free end, and unwind it while keeping the string taut. The pencil draws an involute, and the can is its base circle. Christiaan Huygens studied the curve while thinking about clocks for ships. In the following century Leonhard Euler proposed it for gear teeth, and the profile is sometimes credited to him.

The useful fact is that the taut string always meets the curve it draws at a right angle. Now set two base circles side by side and imagine one string leaving the first and winding onto the second, like a crossed belt between pulleys. Two involute flanks in mesh touch at a point that travels along exactly that straight line, the common tangent of the two base circles. Because the line never moves, it always cuts the line of centers at the same point, so the law of gearing holds through the whole engagement. A pair of involute gears is a belt drive whose belt has been replaced by tooth contact.

Here is the forgiveness. Push the two shafts slightly apart and neither base circle changes, because each belongs to its own gear. The common tangent tilts a little, but it is still a straight line and it still divides the center distance in the same proportion. The speed ratio is untouched. What changes is the angle of that line and the play between the teeth. Spread a 20 degree pair by 1.5 percent and the working pressure angle rises to 22.21 degrees, because the center distance times the cosine of that angle is fixed by the two base circles. The ratio stays where it was.

Cycloidal teeth offer no such grace. A trade history of gear tooth forms carries its verdict in its own headline: their wheel centers must be fixed with as much care as the teeth are shaped.

For anyone drilling shaft holes in plywood or printing a gearbox frame, this is the property that matters most. A drill that wanders costs some backlash. It does not cost the ratio.

Reading a gear: module, pitch and pressure angle

Tooth size is set by one number. The module is the pitch diameter divided by the number of teeth, in millimeters, so a module 2 gear with 20 teeth has a 40 mm pitch circle. The inch system turns the fraction over: diametral pitch is teeth per inch of pitch diameter, and the module equals 25.4 divided by it, which makes 8 diametral pitch a module of 3.175 mm. Two gears mesh correctly only when their pitch and their pressure angle agree. An international standard, ISO 54, specifies the module values, and the preferred values listed on Wikipedia include 1, 1.25, 1.5, 2, 2.5 and 3.

The rest of a standard tooth hangs off the module. The addendum, the height above the pitch circle, is one module. The dedendum below it is 1.25 modules, which leaves a clearance of 0.25 module under the tip of the mating tooth. So the tip diameter is the pitch diameter plus two modules, the root diameter is the pitch diameter less 2.5 modules, and the base diameter is the pitch diameter times the cosine of the pressure angle. The reference shape behind these proportions is a rack, a gear of infinite diameter whose flanks have become straight lines, and ISO 53 defines the standard basic rack. The older 14.5 degree system used a shallower clearance, 0.157 module.

Take module 2 with 20 and 40 teeth at 20 degrees, the pair the generator opens with. The pitch diameters are 40 and 80 mm, the tips 44 and 84 mm, the roots 35 and 75 mm, and the base circles 37.588 and 75.175 mm. The shafts sit half the sum of the pitch diameters apart, which is 60 mm. One tooth and one space together occupy 6.283 mm of the pitch circle, so a tooth with no backlash is 3.142 mm thick there, and with the default 0.1 mm of backlash shared between the gears it is 3.092 mm. The contact ratio comes to 1.64.

The pressure angle is the tilt of the line of action away from the common tangent of the two pitch circles. A steeper angle gives a stronger tooth and lets smaller gears be cut, at the price of a larger share of the tooth force pushing the shafts apart. That share grows with the tangent of the angle, about 26 percent of the driving force at 14.5 degrees and 36 percent at 20.

History settled the trade. One account records that 14.5 degrees was universally adopted as the involute standard by 1913, while many gears were already being cut at 20 degrees for stronger teeth, and it quotes Wilfred Lewis recommending a 20 degree system in 1900. Today most stock gears are 20 degrees, and course notes file 14.5 under old.

Small pinions and the undercut limit

An involute exists only outside its base circle. On a large gear that is no hardship, and from 42 teeth up at 20 degrees the root circle is the larger of the two, so the whole flank is involute. On a small gear the base circle swells past the root. KHK’s 12-tooth example has a base diameter of 33.829 mm and a root diameter of 28.500 mm, which leaves the bottom of every flank with no involute to follow.

A hob is in effect a rack, and when a rack cuts such a gear its corner digs into the foot of the tooth it has just formed. That is undercut. The limit for a rack-type cutter is two divided by the square of the sine of the pressure angle, which gives 17 teeth at 20 degrees. The same formula gives about 32 at 14.5 degrees and about 11 at 25, one more reason the shallow angle lost.

Undercut leaves the tooth weaker at its base, though a trade column on the subject notes that a gear of 16 teeth or fewer can still be usable if its strength and contact ratio meet the needs of the design. The German Wikipedia article on undercut names a practical floor: the theoretical limit is 17 teeth, but undercut matters in practice only below 14. The usual cure is profile shift, cutting the tooth with the tool pulled outward. A rack is also the worst case. Against a real mate of 40 teeth, the test for two gears in mesh passes a pinion of 15, and against 50 teeth it asks for 16.

The generator neither undercuts nor shifts. Below the base circle it draws a straight radial flank and a round fillet, the part-involute, part-straight tooth of the textbooks, then measures whether the mating tips clear it and says so.

Backlash, and how many teeth share the load

Backlash is the amount by which a tooth space exceeds the thickness of the tooth that enters it. Every drive is given a little, to stop the teeth binding when the parts or their mounting are slightly off and to leave room for lubricant. It can come from spreading the centers or from thinning the teeth, and thinning is by far the commoner method. The generator thins each gear by half the figure entered. Measured square to the flanks the gap is smaller by the cosine of the pressure angle, so 0.1 mm on the pitch circle is 0.094 mm between 20 degree teeth.

Contact ratio is the average number of tooth pairs in mesh: the length of the path of contact divided by the base pitch. At exactly one, a pair lets go at the instant the next takes hold. Design notes ask for at least 1.2. At 1.64 the example pair always has one pair engaged and has two for 64 percent of the time.

From outline to a real gear

Printed gears work, within limits that are mostly thermal. One peer-reviewed study printed 12-tooth, module 4.5 gears in PLA, annealed PLA and ABS and ran them at about 1.2 newton meters and 178 rpm. ABS wore least, and annealing improved PLA only modestly. The same paper notes that polymer gears commonly fail by fatigue or by sudden melting. It also cites earlier work, by authors the OpenAlex catalogue places at the University of Warwick, in which gears printed in Nylon 618 outperformed injection-molded nylon 66 at low to medium torque.

A laser removes a strip of material, the kerf, centered on the drawn line. One cutting service lists an average kerf of 0.2 mm for its 3 mm clear acrylic and puts its accuracy at about 0.13 mm either way, and a 2025 study of 6 mm wood, plywood and fiberboard held its kerfs between 0.29 and 0.35 mm. Half the kerf comes off each flank. Every tooth is therefore 0.2 mm thinner than drawn, and a pair gains about 0.4 mm of backlash nobody asked for. Offset the outline before cutting, or enter zero backlash and let the beam supply it.

A milling cutter leaves the opposite mark. The tool is round and it spins, so it cannot cut an inside corner tighter than its own radius, and the root of a milled gear can be no sharper than the cutter. The generator’s root fillet is the clearance times the module, 0.5 mm at module 2, which a cutter 1 mm across can just follow.

Several things the method does not do. It draws spur gears only, not helical, bevel or worm teeth. It applies no profile shift, so a small pinion gets a warning rather than a corrected tooth. Its root fillet is a plain arc, an approximation of the trochoid a real cutter sweeps out. And it rates nothing: bending stress and surface durability are separate calculations that depend on material, speed and face width.

The involute’s indifference to center distance is why a gear drawn in a browser and cut from plywood can run at all. It forgives the drill. It does not forgive a pinion below the limit, or a kerf nobody measured.