Hypotrochoid

A hypotrochoid is a roulette traced by a point attached to a circle of radius $r$ rolling around the inside of a fixed circle of radius $R$, where the tracing point sits at distance $d$ from the center of the rolling circle. It is a vector-valued periodic function of the rotation angle $\theta$, and it is the same family of curve behind the classic Spirograph toy — the ratio of $R$ to $r$ controls how many "petals" the curve makes before it closes back on itself, and $d$ controls how tightly the petals loop. Two well-known special cases fall out of this same equation: when $d = r$ the curve becomes a hypocycloid, and when $r = \frac{R}{2}$ with $d \ne r$ it traces a straight-line ellipse degenerate case. The parametric equations are:

$x(\theta) = (R - r)\cos\theta + d\cos\left(\frac{R - r}{r}\theta\right)$

$y(\theta) = (R - r)\sin\theta - d\sin\left(\frac{R - r}{r}\theta\right)$