The conjugate locus on convex surfaces

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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.
Original languageEnglish
Article number0
Pages (from-to)241-254
Number of pages14
JournalGeometriae Dedicata
Early online date22 Jun 2018
Publication statusPublished - 1 Jun 2019


  • geodesic
  • conjugate locus
  • Jacobi field
  • rotation index
  • evolute
  • convex


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