Symmedians as Hyperbolic Barycenters
Differential Geometry
2025-03-21 v1 Metric Geometry
Abstract
The symmedian point of a triangle enjoys several geometric and optimality properties, which also serve to define it. We develop a new dynamical coordinatization of the symmedian, which naturally generalizes to other ideal hyperbolic polygons beyond triangles. We prove that in general this point still satisfies analogous geometric and optimality properties to those of the symmedian, making it into a hyperbolic barycenter. We initiate a study of moduli spaces of ideal polygons with fixed hyperbolic barycenter, and of some additional optimality properties of this point for harmonic (and sufficiently regular) ideal polygons.
Cite
@article{arxiv.2311.14194,
title = {Symmedians as Hyperbolic Barycenters},
author = {Maxim Arnold and Carlos E. Arreche},
journal= {arXiv preprint arXiv:2311.14194},
year = {2025}
}
Comments
Submitted -- not yet peer reviewed