A hypotrochoid is a roulette traced by a point attached to acircle of radius r rolling around the inside of a fixed circle of radius $R$, where the point is a distance from the center of the interior circle.The parametric equations for a hypotrochoid are:
\[x=(R-r)\cos\theta+d\cos\left(\frac{R-r}{r}\theta\right)\]
\[y=(R-r)\sin\theta-d\sin\left(\frac{R-r}{r}\theta\right)\]

Where $\theta$ is the angle formed by the horizontal and the center of the rolling circle (note that these are not polar equations because $\theta$ is not the polar angle).

Special cases include the hypocycloid with $d=r$ and theellipse with $R=2r$

The classic Spirograph toy traces out hypotrochoid andepitrochoid curves
.