Spherical Cycloids Generated by One Cone Rolling on Another

In this Demonstration, we generate a spherical trochoid with a cone that rolls without slipping on another stationary cone. The generated curve is called a spherical cycloid or spherical trochoid.

Let and be the base circles of the stationary and rolling cones, respectively, with radii and . Let be the distance of the generating point to the center of .

A spherical cycloid is traced by a point on the edge of , that is, ; a spherical trochoid is traced if .

A closed curve is obtained if is rational.

Let be the angle between the planes of and . For a spherical hypotrochoid, , and for a spherical epitrochoid, .

In the extreme cases or , we get a planar hypotrochoid or epitrochoid, respectively.