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:

$x\left(\mathrm{\θ}\right)\= \left(R\+s\cdot r\right) \cdot \cos \left(\mathrm{\θ}\right) -s\cdot r \cdot \cos \left(\frac{R\+s\cdot r}{r}\cdot \mathrm{\θ}\right)$

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

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:   $x\left(\mathrm{\θ}\right)\= \left(R\+s\cdot r\right) \cdot \cos \left(\mathrm{\θ}\right) \+s\cdot L\cdot r \cdot \cos \left(\frac{R\+s\cdot r}{r}\cdot \mathrm{\θ}\right)$ $y\left(\mathrm{\theta }\right)\= \left(R\+s\cdot r\right) \cdot \sin \left(\mathrm{\theta }\right) - L\cdot r \cdot \sin \left(\frac{R\+s\cdot r}{r}\cdot \mathrm{\theta }\right)$ $$ When $s\=1$ and $L \ne 1$ the curve is called an epitrochoid; when $s\=-1$ and $L \ne 1$, the curve is called a hypotrochoid.

	Number of cusps
	Let $k \= \frac{R}{r}$. •  If k is an integer, the curve has k cusps. •  If k is a rational number, $k \= \frac{a}{b}$ and k is expressed in simplest terms, then the curve has a cusps. •  If k is an irrational number, then the curve never closes.

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