### Abstract

Let C be a smooth, convex curve on either the sphere S^{2}, the hyperbolic plane H^{2} or the Euclidean plane E^{2} 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 language | English (US) |
---|---|

Pages (from-to) | 305-324 |

Number of pages | 20 |

Journal | Pacific Journal of Mathematics |

Volume | 274 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2015 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Fingerprint Dive into the research topics of 'On curves and polygons with the equiangular chord property'. Together they form a unique fingerprint.

## Cite this

*Pacific Journal of Mathematics*,

*274*(2), 305-324. https://doi.org/10.2140/pjm.2015.274.305