hypocycloid

A hypocycloid is a plane curve traced by a point on the circumference of a circle of radius r as it rolls inside a larger fixed circle of radius R, without slipping. The locus lies within the fixed circle and depends on the ratio R/r. If R/r is a rational number, the curve closes after a finite interval; when R/r is an integer n, the hypocycloid has n cusps.

A convenient parametric description uses θ as the angle through which the line from the center of the fixed circle to the contact point has rotated. The coordinates of the tracing point are

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

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

For integer n = R/r, the curve closes after θ sweeps 0 to 2π and exhibits n cusps. Notable special cases include:

- R = 2r (n = 2): the hypocycloid degenerates to a straight line segment.

- R = 3r (n = 3): the deltoid, a three-cusped curve.

- R = 4r (n = 4): the astroid, a four-cusped curve.

In contrast to epicycloids, which are traced by a point on a circle rolling outside another circle, hypocycloids arise from interior rolling and feature cusp singularities at the contact points. The term reflects this interior rolling construction. Hypocycloids have historical and geometric significance and appear in various contexts in mathematics and design.