Involute curve explained

Andrew Prestridge · 9 November 2026

The involute of a circle is the path of a point on a straight line as that line rolls, without slipping, around the circle. For a spur gear the circle is the base circle. Each working flank is that curve, and the other flank is its mirror.

That construction is why a pair of involute gears keeps a constant speed ratio when the center distance is a little off. The common normal at the point of contact is always tangent to both base circles, so it is a fixed straight line — the line of action — for as long as both flanks are involutes of those circles. Open the centers and the pressure angle rises. The speed ratio does not: it is the ratio of the base circles, which is the ratio of the tooth numbers.

The curve, in coordinates

With base radius r_b and roll angle t in radians, one parametric form is:

x = r_b (cos t + t sin t)
y = r_b (sin t − t cos t)

At a radius r on the flank the local pressure angle is α_r = arccos(r_b / r), and the roll angle is t = tan α_r. The polar angle from the start of the involute out to that point is the involute function:

inv α = tan α − α

with α in radians. Tooth thickness at any radius, the operating pressure angle of a shifted pair, and the transverse contact ratio are all sums and differences of that function. The designer places the flank so that, at the reference pitch radius, the half-angle matches the circular tooth thickness, and so that the normal to the flank is tangent to the base circle.

Base circle and base pitch

The base circle is fixed by the reference pitch circle and the standard pressure angle: d_b = d cos α. The base pitch is the distance from one flank to the next, measured along the line of action:

p_b = π m cos α = π cos α / P_d

Two external gears mesh only if they share that base pitch. Module (or diametral pitch) and pressure angle have to match. Tooth counts do not. See module and diametral pitch.

What the standards call it

ISO 21771 is the geometry document for cylindrical involute gears: reference, base, and tip diameters, profile shift, and contact ratio. ISO 53 is the basic rack those teeth are generated from. AGMA practice uses the same involute. The argument on a mixed drawing is usually the unit system and the tooth-proportion callout, not a different curve.

Inside the base circle an involute does not exist. The line you see there on a small pinion is the root fillet. A hob cuts that fillet as a trochoid of its tip radius; this tool draws the circular fillet you type in. If the hob would also have cut away part of the involute, the undercut check says so. The contact ratio reported for an undercut pinion is the full-involute value, and it is optimistic.

Contact ratio, briefly

The transverse contact ratio is the length of the path of contact divided by the base pitch:

ε_α = ( √(r_a1² − r_b1²) + √(r_a2² − r_b2²) − a sin α_w ) / (π m cos α)

Here a is the operating center distance and α_w is the operating pressure angle. For two standard 20-tooth, 20° full-depth gears with no shift, that comes out to about 1.56. The designer uses the tip radii after any tip shortening from profile shift, so the number matches the mesh on the screen.