Tripod configurations of curves

TL;DR

This paper proves the existence of tripod configurations for all $C^2$ closed plane curves, extending previous results and exploring their analogues in spherical, hyperbolic geometries, and for polygons.

Contribution

It generalizes the existence of tripod configurations to all $C^2$ closed curves and extends these concepts to spherical and hyperbolic geometries.

Findings

Existence of generalized tripod configurations for all $C^2$ closed plane curves.

Extension of tripod configurations to spherical and hyperbolic geometries.

Discussion of tripod configurations in regular polygons.

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 two results, answering a conjecture of Tabachnikov on the existence of tripod configurations for all $C^2$ 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.