The Conjugate Locus on the Triaxial Ellipsoid
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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].
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Contributed by: Thomas Waters (April 2014)
Open content licensed under CC BY-NC-SA
Snapshots
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.
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