Details

Let be the radius of the base circle centered at , let be the radius of the rolling circle, and let be the angle between the - plane and the plane of the rolling circle.

Let be the angular displacement of the rolling circle along the edge of the base circle. Since the rolling circle rolls without sliding, its angular displacement around its center is .

is a point on a copy of the rolling circle centered at in the - plane. First rotate this circle by around the axis to obtain:

.

Now translate the circle over a distance along the axis to get:

.

Finally, rotate this circle by an angle around the axis:

to obtain the parametric equations of the spherical cycloid:

,

,

.

Reference

[1] H. M. Jeffery, "On Spherical Cycloidal and Trochoidal Curves," The Quarterly Journal of Pure and Applied Mathematics, 19(73), 1882 pp. 45–66. gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN600494829_0019%7CLOG_0012.