What is "hypotrochoid"

Hypotrochoid \Hy`po*tro"choid\, n. [Pref. hypo- + trochoid.] (Geom.) A curve, traced by a point in the radius, or radius produced, of a circle which rolls upon the concave side of a fixed circle. See Hypocycloid, Epicycloid, and Trochoid.

n. (context geometry English) A geometric curve traced by a fixed point on the radius line outside one circle which rotates inside the perimeter of another circle.

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:

$$x (\theta) = (R - r)\cos\theta + d\cos\left({R - r \over r}\theta\right)$$

$$y (\theta) = (R - r)\sin\theta - d\sin\left({R - r \over r}\theta\right).$$

Where θ is the angle formed by the horizontal and the center of the rolling circle (note that these are not polar equations because θ 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.