In geometry, a hypocycloid is a special plane curve, a roulette, generated by the trace of a fixed point on a small circle that rolls within a larger circle. It is comparable to the cycloid but instead of the circle rolling along a line, it rolls within a circle.

The ratio of the radius of the larger circle to the radius of the smaller circle determines the number of cusps of the curve. For example if the ratio is 3:1 the curve will have three cusps and it will be a deltoid.

Such curves can be drawn with the Spirograph toy.

A hypocycloid with n + 1 cusps can be defined by the following pair of parametric equations:

The hypocycloid is a special kind of hypotrochoid.

A hypocycloid and its evolute are similar.[1]

A hypocycloid curve with four cusps is known as an astroid.

See also: cycloid, epicycloid.