Involute Gears Explained

Every gear generated on this site uses an involute tooth profile — the shape traced by the end of a taut string as it unwinds from a circle (the base circle). It's not an arbitrary curve: it's the one shape that keeps the contact point between two meshing teeth moving along a straight line (the line of action) throughout the mesh, which keeps the velocity ratio between the gears constant even if the center distance is slightly off from nominal. That tolerance to small assembly error is the main reason the involute profile displaced older cycloidal and other tooth forms for general-purpose gearing over a century ago.

Mathematically, a point on the involute at radius r (measured from the gear center) sits at a polar angle offset from the tooth's reference line by the involute function:

inv(α) = tan(α) − α, where α = acos(base radius / r).

That's exactly the function this site's gear engine (gearProfile() in the generator source) evaluates at each radial step from the base circle to the tooth tip to build the flank curve you see in every SVG diagram on this site.

Base circle vs. root circle

A common point of confusion: the involute curve only exists outside the base circle (radius = pitch radius × cos(pressure angle)). Since the dedendum (root depth) is usually deeper than that, the flank below the base circle isn't an involute at all — it's a radial line blended into the root with a fillet, which is exactly what you'll see if you zoom into the root of any gear's technical diagram on this site.

See it in the catalog

Compare the flank curvature of a 10-tooth and a 60-tooth gear at the same module — more teeth means a larger base circle relative to the tooth, so the flank looks straighter (a gear with an infinite tooth count, i.e. a rack, has perfectly straight involute flanks). Related: Pressure Angle Explained covers how the 20° angle used throughout this catalog sets the base circle's size relative to the pitch circle.