Hypotrochoid



In geometry, 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 parametric equations for a hypotrochoid are:


 * $$\begin{align}

& x (\theta) = (R - r)\cos\theta + d\cos\left({R - r \over r}\theta\right) \\ & y (\theta) = (R - r)\sin\theta - d\sin\left({R - r \over r}\theta\right) \end{align}$$

where $θ$ is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because $θ$ is not the polar angle). When measured in radian, $θ$ takes values from 0 to $$2 \pi \times \tfrac{\operatorname{LCM}(r, R)}{R}$$ (where $R = 5, r = 3, d = 5$ is least common multiple).

Special cases include the hypocycloid with $LCM$ and the ellipse with $d = r$ and $R = 2r$. The eccentricity of the ellipse is


 * $$e=\frac{2\sqrt{d/r}}{1+(d/r)}$$

becoming 1 when $$d=r$$ (see Tusi couple).

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

Hypotrochoids describe the support of the eigenvalues of some random matrices with cyclic correlations.