Hypocycloid
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In geometry, a hypocycloid is a special plane curve, a roulette, 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.
The ratio of the radius of the larger circle to the radius of the smaller circle determines the number of cusps of the curve. For example if the ratio is 3:1 the curve will have three cusps and it will be a deltoid.
Such curves can be drawn with the Spirograph toy.
A hypocycloid with n + 1 cusps can be defined by the following pair of parametric equations:
- <math> x(\theta) = \cos \theta + {1 \over n} \cos n \theta, <math>
- <math> y(\theta) = \sin \theta - {1 \over n} \sin n \theta. <math>
The hypocycloid is a special kind of hypotrochoid.
A hypocycloid and its evolute are similar.[1] (http://mathworld.wolfram.com/HypocycloidEvolute.html)
A hypocycloid curve with four cusps is known as an astroid.
The Pittsburgh Steelers' logo includes three astroids (hypocycloids of four cusps).
See also: cycloid, epicycloid.
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