Epicycloid

An epicycloid is the curve traced by a fixed point on a circle of radius r as it rolls without slipping on the outside of a fixed circle of radius R. It is a special case of a trochoid and is studied in geometry and kinematics for its characteristic looping shapes.

If the tracing point is at a distance d from the center of the rolling circle, its position can be described by the parametric equations

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

y(t) = (R + r) sin t − d sin(((R + r)/r) t),

where t is the rolling parameter. For the common case where the tracing point lies on the circumference of the rolling circle (d = r), the equations simplify to

x(t) = (R + r) cos t − r cos(((R + r)/r) t)

y(t) = (R + r) sin t − r sin(((R + r)/r) t).

A key property is how the shape depends on the ratio R/r. If R/r is an integer n, the epicycloid has n cusps. The case n = 1 (R = r) yields a cardioid, and n = 2 (R = 2r) yields a nephroid. If R/r is irrational, the curve does not close and continues to fill a region densely.

Epicycloids appear in various contexts, including decorative patterns, gear design and epicyclic mechanisms, and handheld drawing toys like the Spirograph. They are related to other cycloidal curves and contribute to the broader study of rolling curves and their geometry.