Epicycloid

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

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.

The epicycloid is a special kind of epitrochoid.

An epicycle with one cusp is a cardioid.

An epicycloid and its evolute are similar.

See also

• Special cases: Cardioid, Nephroid
• Cycloid
• Hypocycloid
• Epitrochoid
• Hypotrochoid
• Spirograph
• Deferent and epicycle

References

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications.

External links

• Epicycloid, MathWorld
• "Epicycloid" by Michael Ford, The Wolfram Demonstrations Project, 2007