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**Parametric Cartesian equation: x = (***a + b*)cos(*t*) - *b*cos((*a/b* + 1)*t*), y = (*a + b*)sin(*t*) - *b*sin((*a/b* + 1)*t*)

There are four curves which are closely related. These are the epicycloid, the epitrochoid, the hypocycloid and the hypotrochoid and they are traced by a point *P* on a circle of radius *b* which rolls round a fixed circle of radius *a*.

For the epicycloid, shown above in clarity-reducing ASCII, the circle of radius *b* rolls on the outside of the circle of radius *a*. The point *P* is on the circumference of the circle of radius *b*. For the example drawn here *a* = 8 and *b* = 5.

**Special cases**

*a* = *b*, a cardioid is obtained
*a* = 2*b*, a nephroid is obtained
*a* = (*n* - 1)*b* where *n* is an integer, then the length of the epicycloid is 8*nb* and its area is *b*^{2}(*n*^{2} + n)

Also, the

evolute of an

epicycloid is a similar

epicycloid.