Epicycloid

An epicycloid is the curve traced by a fixed point on the circumference of a circle as it rolls, without slipping, around the outside of a fixed circle. It is the mirror-image counterpart of the hypotrochoid family above (traced by a circle rolling on the inside), and the cardioid is its simplest special case — an epicycloid where the rolling and fixed circles are exactly the same size. For a fixed circle of radius $R$ and a rolling circle of radius $r$, the ratio $k=R/r$ determines how many cusps the curve makes before it closes back on itself: an integer $k$ always produces exactly $k$ cusps. Setting $r=1$, the parametric equations (in terms of $k$) are:

$x(\theta) = (k+1)\cos\theta - \cos((k+1)\theta)$

$y(\theta) = (k+1)\sin\theta - \sin((k+1)\theta)$