Hypotrochoid

A hypotrochoid is a plane curve traced by a point attached to a circle of radius r that rolls without slipping inside a larger fixed circle of radius R. The tracing point is at a distance d from the center of the rolling circle. It is a member of the trochoid family and generalizes both the hypocycloid (when d equals r) and the inner circular motion when d equals 0.

Mathematically, if θ is the angle of rotation of the rolling circle, the coordinates of the tracing point are given by:

x(θ) = (R − r) cos θ + d cos(((R − r)/r) θ)

y(θ) = (R − r) sin θ − d sin(((R − r)/r) θ).

These equations describe the combination of the moving circle’s center along a circular path of radius R − r and the offset of the tracing point on the rolling circle.

Special cases and properties: when d = 0, the locus is a circle of radius R − r. When d = r, the curve reduces to a hypocycloid. If R/r is rational, the hypotrochoid is closed and retraces itself after a finite number of revolutions; if R/r is irrational, the curve does not close and, in effect, does not repeat exactly.

Variations in the parameters produce a wide range of shapes, from smooth rounded figures to multi-cusped or looped patterns. Hypotrochoids are well known for decorative designs produced by the Spirograph toy, and they have applications in art and computer graphics as examples of richly parameterized planar curves.