Tripod configurations of curves

Tripod configurations of plane curves, formed by certain triples of normal lines coinciding at a point, were introduced by Tabachnikov, who showed that C2C2 closed convex curves possess at least two tripod configurations. Later, Kao and Wang established the existence of tripod configurations for C2C2 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.