Epicycloid (idea)

The epicycloid is the path traced by a point on the edge of a circle as it rolls without slipping along the edge of another circle. The two circles are not required to be the same size:

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With the above setup, let t be the angle that circle 1 (having radius R1) has rotated around circle 2 (radius R2). Then the parametric equations for the path a point on the edge of circle 1 traces is:

x=(R1+R2)*cos(t*R1/R2)-R1*cos(t)
y=(R1+R2)*sin(t*R1/R2)-R1*sin(t)

Note that, if R1=R2, then the shape is a cardioid. (In the above setup, I used R1=25chars, R2=24chars.)