Epicycloid

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In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — called epicycle — which rolls around without slipping around a fixed circle. It is a particular kind of roulette.

The red curve is an epicycloid traced as the small circle (radius r = 1) rolls around the outside of the large circle (radius R = 3).
The red curve is an epicycloid traced as the small circle (radius r = 1) rolls around the outside of the large circle (radius R = 3).

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:

x(\theta) = r (k+1) \left( \cos \theta - \frac{\cos((k+1)\theta)}{k+1} \right)
y(\theta) = r (k+1) \left( \sin \theta - \frac{\sin((k+1)\theta)}{k+1} \right)

If k is an integer, then the curve is closed, and has k cusps (i.e., sharp corners, where the curve is not differentiable).

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.

epicycloid examples

k = 1

k = 2

k = 3

k = 4

k = 2.1 = 21/10

k = 3.8 = 19/5

k = 5.5 = 11/2

k = 7.2 = 36/5

The epicycloid is a special kind of epitrochoid.

An epicycle with one cusp is a cardioid.

An epicycloid and its evolute are similar.[1]

[edit] See also

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