Epitrochoids

An epitrochoid is a plane curve traced by a point attached to a circle of radius r that rolls without slipping around the outside of a fixed circle of radius R. The tracing point is at a distance d from the center of the rolling circle. As the rolling circle turns around the fixed circle, the combination of rotation and translation generates the locus.

The curve can be described parametrically 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),

where θ is the angle through which the rolling circle has rotated. A useful form uses k = R/r, giving x(θ) = r(k + 1) cos θ − d cos((k + 1) θ), y(θ) = r(k + 1) sin θ − d sin((k + 1) θ).

Special cases and relations: If d = r, the locus is an epicycloid. If d = 0, the tracing point is at the center of the rolling circle, yielding a circle of radius R + r. The curve is bounded within an annulus whose radii range between |R + r − d| and R + r + d. If R/r is a rational number, the epitrochoid is closed after a finite number of revolutions; if irrational, the tracing is not closed and is dense in that annulus.

Epitrochoids encompass many familiar shapes and are a generalization of trochoid curves. They are popularly realized in spirograph drawing toys and have applications in mechanical design and geometric art.