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: 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 (see Tusi couple). The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.