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 point is a distance d from the center of the interior circle.

The red curve is a hypotrochoid drawn as the smaller black circle rolls around inside the larger blue circle (parameters are R = 5, r = 3, d = 5).

The parametric equations for a hypotrochoid are:^[1]

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

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

where ${\displaystyle \theta }$ is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because ${\displaystyle \theta }$ is not the polar angle). When measured in radian, ${\displaystyle \theta }$ takes values from ${\displaystyle 0}$ to ${\displaystyle 2\pi \times {\frac {\operatorname {LCM} (r,R)}{R}}}$where LCM is least common multiple.

Special cases include the hypocycloid with d = r is a line or flat ellipse and the ellipse with R = 2r and d > r or d < r (d is not equal to r).^[2] (see Tusi couple).

The ellipse (drawn in red) may be expressed as a special case of the hypotrochoid, with R = 2r (Tusi couple); here R = 10, r = 5, d = 1.

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

Hypotrochoids describe the support of the eigenvalues of some random matrices with cyclic correlations^[3]