11193

Spherical Cycloid

This Demonstration simulates the generation of a spherical cycloid by a point on the edge of a circle rolling without sliding along the edge of another circle (the base circle) on the same sphere (the base sphere).
is the angle between the planes of the base circle and the rolling circle.
creates a spherical hypocycloid and gives a spherical epicycloid.
In the extreme cases of or , we get a planar hypocycloid and epicycloid, respectively.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

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.
    • 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 »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+