Hypotrochoid
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. When measured in radian, takes values from to where LCM is least common multiple.
Special cases include the hypocycloid with d = r is a line or flat ellipse and the ellipse with R = 2r and d > r or d < r..
may be expressed as a special case of the hypotrochoid, with R = 2r ; here R = 10, r = 5, d = 1.
The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.
