If the smaller circle has radius r, and the larger circle has radius R = kr, then the parametric equations for the curve can be given by
If k is an integer, then the curve is closed, and has k cusps (i.e., sharp corners, where the curve is not differentiable). Specially for k=2 the curve is a straight line and the circles are called Cardano circles. Girolamo Cardano was the first to describe these hypocycloids, which had applications in the technology of high-speed printing press.
If k is a rational number, say k = p/q expressed in simplest terms, then the curve has p cusps.
If k is an irrational number, then the curve never closes, and fills the space within the larger circle except for a disk of radius R − r in the center of the larger circle.
A hypocycloid with three cusps is known as a deltoid.
A hypocycloid curve with four cusps is known as an astroid.
The isoptic of a hypocycloid is a hypocycloid.
The Pittsburgh Steelers' logo includes three astroids (hypocycloids of four cusps). In his weekly NFL.com column Tuesday Morning Quarterback, Gregg Easterbrook often refers to the Steelers as the Hypocycloids.
Portland, Oregon's flag features a hypocycloid; an astroid.
The 2007 redesign of The Price is Right's set features astroids on the three main doors and the turntable area