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:[1]
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.[2] The eccentricity of the ellipse is
becoming 1 when (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.[3]