epicycloid: Definition from Answers.com

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).

In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — called an 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 either:

$x (\theta) = (R + r) \cos \theta - r \cos \left( \frac{R + r}{r} \theta \right)$

$y (\theta) = (R + r) \sin \theta - r \sin \left( \frac{R + r}{r} \theta \right),$

or:

$x (\theta) = r (k + 1) \cos \theta - r \cos \left( (k + 1) \theta \right) \,$

$y (\theta) = r (k + 1) \sin \theta - r \sin \left( (k + 1) \theta \right). \,$

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 forms a dense subset of 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]

1 Proof
2 See also
3 References
4 External links

Proof

This article may require cleanup to meet Wikipedia's quality standards. Please improve this article if you can. (December 2008)

We assume that the position of $p$ is what we want to solve, $α$ is the radian from the tangential point to the moving point $p$ , and $θ$ is the radian from the starting point to the tangential point.

Since there is no sliding between the two cycles, then we have that

$\ell_R=\ell_r$

By the definition of radian (which is the rate arc over radius), then we have that

$\ell_R= \theta R, \ell_r=\alpha r$

From the two condition, we get the identity

θ R = α r

By calculating, we get the relation between $α$ and $θ$ , which is

$\alpha =\frac{R}{r} \theta$

From the figure, we see the position of the point $p$ clearly.

$x=\left( R+r \right)\cos \theta -r\cos\left( \theta+\alpha \right) =\left( R+r \right)\cos \theta -r\cos\left( \frac{R+r}{r}\theta \right)$

$y=\left( R+r \right)\sin \theta -r\sin\left( \theta+\alpha \right) =\left( R+r \right)\sin \theta -r\sin\left( \frac{R+r}{r}\theta \right)$

References

J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 161,168–170,175. ISBN 0-486-60288-5.

External links

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

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	epicycle (mathematics)
	epitrochoid
	nephroid (mathematics)

epicycloid

Contents

Proof

See also

References

External links