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arXiv · 1806.00278

The conjugate locus on convex surfaces

Thomas Waters
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Abstract

The conjugate locus of a point on a surface is the envelope of geodesics emanating radially from that point. In this paper we show that the conjugate loci of generic points on convex surfaces satisfy a simple relationship between the rotation index and the number of cusps. As a consequence we prove the `vierspitzensatz': the conjugate locus of a generic point on a convex surface must have at least four cusps. Along the way we prove certain results about evolutes in the plane and geodesic curvature. (Note: this is a corrected version of the original paper, see comment on page 5 and Appendix B).

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Thomas Waters. 2018-06-01. The conjugate locus on convex surfaces. https://arxiv.org/abs/1806.00278

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