On curves and polygons with the equiangular chord property

Tarik Aougab, Xidian Sun, Serge Tabachnikov, Yuwen Wang

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

Let C be a smooth, convex curve on either the sphere S2, the hyperbolic plane H2 or the Euclidean plane E2 with the following property: there exists α and parametrizations x(t) and y(t) of C such that, for each t, the angle between the chord connecting x(t) to y(t) and C is α at both ends. Assuming that C is not a circle, E. Gutkin completely characterized the angles α for which such a curve exists in the Euclidean case. We study the infinitesimal version of this problem in the context of the other two constant curvature geometries, and in particular, we provide a complete characterization of the angles α for which there exists a nontrivial infinitesimal deformation of a circle through such curves with corresponding angle α. We also consider a discrete version of this property for Euclidean polygons, and in this case, we give a complete description of all nontrivial solutions.

Original languageEnglish (US)
Pages (from-to)305-324
Number of pages20
JournalPacific Journal of Mathematics
Volume274
Issue number2
DOIs
StatePublished - Jan 1 2015

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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