On the Last Geometric Statement of Jacobi

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 R³ (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 −p (odd caustics). This is an extension of a statement usually attributed to Jacobi.

Keywords:

• Conjugate locus
• caustic
• ellipsoid