Epicycloid

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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]

Contents

[edit] Proof

 This article may require cleanup to meet Wikipedia's quality standards. Please improve this article if you can. (December 2008)
Pf1.jpg

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

θR = αr

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)

[edit] See also

[edit] References

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

[edit] External links

Retrieved from "http://en.wikipedia.org/wiki/Epicycloid"