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In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. It is comparable to the cycloid but instead of the circle rolling along a line, it rolls within a circle.
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 either:
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 between the larger circle and a circle of radius R − 2r.
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.
Hypocycloids in popular culture
The Pittsburgh Steelers' logo, which is based on the Steelmark, 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.
The first Drew Carey season of The Price Is Right's set features astroids on the three main doors, giant price tag, and the turntable area. The astroids on the doors and turntable were removed when the show switched to high definition broadcasts starting in 2008, and only the giant price tag prop still features them today. 
- Special cases: Astroid, Deltoid
- List of periodic functions
- Flag of Portland, Oregon, which features a hypocycloid
- J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 168,171–173. ISBN 0-486-60288-5.
- Hazewinkel, Michiel, ed. (2001), "Hypocycloid", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- O'Connor, John J.; Robertson, Edmund F., "Hypocycloid", MacTutor History of Mathematics archive, University of St Andrews.
- Plot Hypcycloid — GeoFun
- Iterative demonstration showing the brachistochrone property of Hypocycloid