cycloid gear

is a form of toothed profile is based the epicycloid and hypocycloid curves, which are the curves generated by a circle rolling round the outside (respectively inside) of another circle.

When two toothed gears mesh, an imaginary circle, the pitch circle, can be drawn around the centre of either gear through the point at which their teeth make contact. The curves of the teeth outside the pitch circle are known as the addenda, and the curves of the tooth spaces inside the pitch circle are known as the dedenda. An addendum of one gear rests inside a dedendum of the other gear.

In cycloidal gears, the addenda of the wheel teeth are convex epi-cycloidal and the dedenda of the pinion are concave hypocycloidal curves generated by the same generating circle. This ensures that the motion of one gear is transferred to the other at locally constant angular velocity.

epicycloid The red curve is an epicycloid traced as the small circle (radius r = 1) rolls around the outside of the large circle (radius R = 3).

hypocycloid The red curve is a hypocycloid traced as the smaller black circle rolls around inside the larger black circle (parameters are R=3.0, r=1.0, so k=3).