TY - JOUR

T1 - On curves and polygons with the equiangular chord property

AU - Aougab, Tarik

AU - Sun, Xidian

AU - Tabachnikov, Serge

AU - Wang, Yuwen

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

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U2 - 10.2140/pjm.2015.274.305

DO - 10.2140/pjm.2015.274.305

M3 - Article

AN - SCOPUS:84929237638

SN - 0030-8730

VL - 274

SP - 305

EP - 324

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

IS - 2

ER -