Epitrochoid

An epitrochoid is a plane curve traced by a point attached to a circle that rolls around the outside of a fixed circle. If a circle of radius r rolls externally around a fixed circle of radius R, and a point at a distance d from the rolling circle’s center moves with it, the locus of the point is an epitrochoid.

The curve can be described parametrically by:

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

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

where θ is the rotation angle of the rolling circle about the fixed circle. Special case: when d = r, the tracing point lies on the rolling circle’s circumference and the curve is an epicycloid. When d = 0, the path is a circle of radius R + r.

Epitrochoids can produce a wide family of shapes, ranging from smooth rosettes to cusped forms, depending on the ratio R/r and the offset d. The curve is closed if (R + r)/r is a rational number; otherwise it does not exactly repeat.

These curves are studied in geometry and are used in decorative design and artistic rendering, and they appear in devices such as Spirograph toys and in certain mechanical cam problems.