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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.
Derived curves 
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.
See also 
- 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.