Experimental Mathematics

On the Last Geometric Statement of Jacobi

R. Sinclair

Abstract

Based upon numerical experimentation, we claim that all caustics from any nonumbilical (nonpolar for ellipsoids of revolution) point p on any ellipsoid embedded in {\small $\mathbb{R}^3$} (except the 2-sphere) have exactly four cusps, all of which are on lines of curvature (meridians and parallels for ellipsoids of revolution) intersecting either p (even caustics) or {\small $-p$} (odd caustics). This is an extension of a statement usually attributed to Jacobi.

Article information

Source
Experiment. Math. Volume 12, Number 4 (2003), 477-486.

Dates
First available in Project Euclid: 18 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.em/1087568023

Mathematical Reviews number (MathSciNet)
MR2043997

Zentralblatt MATH identifier
1073.53007

Subjects
Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]
Secondary: 53-04: Explicit machine computation and programs (not the theory of computation or programming)

Keywords
Conjugate locus caustic ellipsoid

Citation

Sinclair, R. On the Last Geometric Statement of Jacobi. Experiment. Math. 12 (2003), no. 4, 477--486. https://projecteuclid.org/euclid.em/1087568023.


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