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{aligned}&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{aligned}}

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 $LCM$ is least common multiple).

Special cases include the hypocycloid with $d = r$ and the ellipse with $R = 2 r$ and $d \neq r$ . 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.

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