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 is the angle formed by the horizontal and the center of the rolling circle (note that these are not polar equations because is not the polar angle).Special cases include the hypocycloid with d = r and the ellipse with R = 2r.The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.