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:


xθ= R+sr  cosθ sr  cosR+srrθ 

yθ= R+sr  sinθ r  sinR+srrθ 


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:


xθ= R+sr  cosθ +sLr  cosR+srrθ 

yθ= R+sr  sinθ  Lr  sinR+srrθ 

When s=1 and L  1 the curve is called an epitrochoid; when s=1 and L  1, the curve is called a hypotrochoid.

Number of cusps

Let k  = Rr.


If k is an integer, the curve has k cusps.


If k is a rational number, k = ab 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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