The Hypotrochoid

Introduction:

The hypotrochoid is similar to the epitrochoid, except that the smaller circle rolls on the inside edge of the larger one. It has as special cases the ellipse (a = 2b) and the hypocycloid (h = b). In general, it has a/b loops.

Definition:

Geometrically, the hypotrochoid is traced by a point P, attached at a distance h to the center of a small circle of radius b as it rolls along the inside edge of a larger circle of radius a.

Properties:

Velocity and Acceleration:

The Tangent Vector:

Arc Length:

Curvature: