Epicycloid and Hypocycloid - Maple Help

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Epicycloid and Hypocycloid

Main Concept

An epicycloid is a plane curve created by tracing a chosen point on the edge of a circle of radius r rolling on the outside of a circle of radius R. A hypocycloid is obtained similarly except that the circle of radius r rolls on the inside of the circle of radius R.

The parametric equations for the epicycloid and hypocycloid are:

where $s=1$ for the epicycloid and $s=-1$ for the hypocycloid.

 Epitrochoid and hypotrochoid Two related curves result when we include another parameter, L, which represents the ratio of pen length to the radius of the circle:    When $s=1$ and  the curve is called an epitrochoid; when $s=-1$ and , the curve is called a hypotrochoid.

Number of cusps

Let .

 • If k is an integer, the curve has k cusps.
 • If k is a rational number,  and k is expressed in simplest terms, then the curve has a cusps.
 • If k is an irrational number, then the curve never closes.

 Fixed circle radius (R) = Rolling circle radius (r) = Start End Ratio of Pen length/radius Animation speed



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