Tripod configurations of curves

Abstract

Tripod configurations of plane curves, formed by certain triples of normal lines coinciding at a point, were introduced by Tabachnikov, who showed that C 2 closed convex curves possess at least two tripod configurations. Later, Kao and Wang established the existence of tripod configurations for C 2 closed locally convex curves. In this paper we generalize these results, answering a conjecture of Tabachnikov on the existence of tripod configurations for all closed plane curves by proving existence for a generalized notion of tripod configuration. We then demonstrate the existence of the natural extensions of these tripod configurations to the spherical and hyperbolic geometries for a certain class of convex curves, and discuss an analogue of the problem for regular plane polygons.