Re-Orient a Sphere with Two Straight Rolls

It is well-known that a sphere can be rolled in the horizontal plane to any orientation by three sequentially orthogonal rolls. This Demonstration shows that only two straight rolls are necessary. The desired orientation, shown in red, is specified by moving the north pole to a desired latitude and longitude , then twisting about this pole by . Rolling from the initial orientation, shown in green, along the blue and red lines brings the sphere to the desired orientation.


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


The composite rotation of about the axis, about the axis, and about the axis can be reproduced by two straight rolls in the horizontal plane. For the unit sphere, a roll of at an angle with the axis, followed by a roll of length at an angle with the axis produces the same rotation.
Any desired rotation can be specified by three sequentially orthogonal rotations:
A rotation on the plane of length , making angle with the axis produces the rotation
The two straight rolls reproduce the desired rotation:
[1] J. M. Hammersley, "7. Oxford Commemoration Ball," in Probability, Statistics and Analysis: London Mathematical Society Lecture Note Series, Number 79, ed. J. F. C. Kingman and G. E. H. Reuter, Cambridge University Press, 1983.
    • 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.

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 © 2018 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+