An epicycloid is the path traced by a point on a circle of radius r that rolls on the outs...

An epicycloid is the path traced by a point on a circle of radius r that rolls on the outside of a fixed circle of radius R. By placing the moving circle initially so that it and the fixed circle are initially tangent at (R, 0) and by having it roll counterclockwise in the xy-plane, show that the epicycloid is parametrized by

where t is the angle ∠PCT shown in Figure 1.

Figure 1

An epicycloid is generated by a point P on a circle that rolls on the outside of another