[Symédianes comme barycentres hyperboliques]
Le point symédiane d’un triangle possède plusieurs propriétés géométriques d’optimalité, qui peuvent servir à la définir. Nous développons une nouvelle définition dynamique du point symédiane, qui se généralise naturellement à d’autres polygones idéaux hyperboliques, au-delà des triangles. Nous prouvons que, de manière générale, ce point satisfait toujours des propriétés géométriques d’optimalité analogues à celles du point symédiane, qui en font un barycentre hyperbolique. Nous entamons une étude des espaces de modules des polygones idéaux dont le barycentre hyperbolique est fixe, ainsi que de certaines propriétés d’optimalité supplémentaires pour les polygones idéaux harmoniques (et suffisamment réguliers).
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.
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Keywords: symmedian point, Hyperbolic barycenter, harmonic polygon, ideal hyperbolic polygon
Mots-clés : Barycentre hyperbolique, polygone harmonique, polygone hyperbolique idéal
Maxim Arnold 1 ; Carlos E. Arreche 1
CC-BY 4.0
@article{CRMATH_2024__362_G12_1743_0,
author = {Maxim Arnold and Carlos E. Arreche},
title = {Symmedians as {Hyperbolic} {Barycenters}},
journal = {Comptes Rendus. Math\'ematique},
pages = {1743--1762},
publisher = {Acad\'emie des sciences, Paris},
volume = {362},
year = {2024},
doi = {10.5802/crmath.677},
language = {en},
}
Maxim Arnold; Carlos E. Arreche. Symmedians as Hyperbolic Barycenters. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1743-1762. doi : 10.5802/crmath.677. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.677/
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