[[Image:HypotrochoidOutThreeFifths.gif|thumb|400px| 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 ).]] 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: 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 (where LCM is least common multiple). Special cases include the hypocycloid with d = r and the ellipse with R = 2r and d ≠ r . The eccentricity of the ellipse is becoming 1 when d=r (see Tusi couple). [[Image:Ellipse as hypotrochoid.gif|right|400px|thumb|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.