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).

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.