11428

# Spherical Trochoid

This Demonstration simulates the generation of a spherical trochoid by a point attached to a circle rolling without sliding along the edge of another circle (the base circle) on the same sphere (the base sphere).
A spherical cycloid is traced by a point on the rolling circle's edge; a spherical trochoid is drawn by a point attached to the circle at a distance greater than or less than its radius. A spherical trochoid becomes a spherical cycloid if the distance of the generating point to the rolling circle's center is equal to its radius.
is the angle between the planes of the base circle and the rolling circle.
For a spherical hypotrochoid, , and for a spherical epitrochoid, .
In the extreme cases, or , we get a planar hypotrochoid or epitrochoid, respectively.

### DETAILS

Let be the radius of the base circle centered at , the radius of the rolling circle and the distance of the generating point to its center; is 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 .
The point on a copy of the rolling circle centered at in the - plane and at a distance from its center is:
.
First rotate this circle by around the axis:
.
Now translate the circle over a distance along the axis to get:
.
Finally, rotate this circle by an angle around the axis:
.
This gives the parametric equation of the spherical trochoid:
The spherical trochoid is on a sphere with center at and radius .
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.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.