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]
A hypocycloid curve with four cusps is known as an astroid.
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.
See also: cycloid, epicycloid.af:Hiposikloïed de:Hypozykloide fr:hypocycloïde it:ipocicloide nl:Cycloïdes#Hypocycloïde pl:Hipocykloida zh:圆内螺线