The conjugate locus on convex surfaces

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...

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Veröffentlicht in:	Geometriae dedicata 2019-06, Vol.200 (1), p.241-254
1. Verfasser:	Waters, Thomas
Format:	Artikel
Sprache:	eng
Schlagworte:	Algebraic Geometry Conjugates Convex and Discrete Geometry Curvature Cusps Differential Geometry Geodesy Hyperbolic Geometry Loci Mathematics Mathematics and Statistics Original Paper Projective Geometry Topology
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description	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 we extend the discussion to the existence of ‘smooth loops’ and geodesic curvature.
doi_str_mv	10.1007/s10711-018-0368-8
format	Article
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subjects	Algebraic Geometry Conjugates Convex and Discrete Geometry Curvature Cusps Differential Geometry Geodesy Hyperbolic Geometry Loci Mathematics Mathematics and Statistics Original Paper Projective Geometry Topology
title	The conjugate locus on convex surfaces
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