10182

# The Conjugate Locus on the Triaxial Ellipsoid

The "last geometric statement of Jacobi" says, among other things, that the conjugate locus of a non-umbilic point on the triaxial ellipsoid has four cusps. This was only recently proved by Itoh and Kiyohara [1].
In this Demonstration we show the locus of conjugate points along a "spray" of unit speed geodesics emanating from a "base point" on the ellipsoid with semiaxes 1, 1.2, and 1.5. You can vary the coordinates of the base point and the number of geodesics in the spray. We also offer two different views (in the second view, the blue sphere is the base point).

### DETAILS

The parameterization of the ellipsoid used is , with , , and .
It can be a little slow to render the image after dragging the and controls. Having said that, Mathematica is solving, for each and , the geodesics equations and the Jacobi equation tens of times, so it is still impressive what Mathematica can do.
Try to find the umbilic points; these are base points where the conjugate locus degenerates to a point (the antipode).
Reference
[1] J. Itoh and K. Kiyohara, "The Cut Loci and the Conjugate Loci on Ellipsoids," Manuscripta Mathematica, 114, 2004 pp. 247–264.

### 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.