Symmedians as Hyperbolic Barycenters
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 stil...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.677/ |
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| author | Arnold, Maxim Arreche, Carlos E. |
| author_facet | Arnold, Maxim Arreche, Carlos E. |
| author_sort | Arnold, Maxim |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-4b42b98e9e034f40911735804781936e |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-4b42b98e9e034f40911735804781936e2025-02-07T11:26:37ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G121743176210.5802/crmath.67710.5802/crmath.677Symmedians as Hyperbolic BarycentersArnold, Maxim0Arreche, Carlos E.1Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, USADepartment of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, USAThe 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.677/symmedian pointHyperbolic barycenterharmonic polygonideal hyperbolic polygon |
| spellingShingle | Arnold, Maxim Arreche, Carlos E. Symmedians as Hyperbolic Barycenters Comptes Rendus. Mathématique symmedian point Hyperbolic barycenter harmonic polygon ideal hyperbolic polygon |
| title | Symmedians as Hyperbolic Barycenters |
| title_full | Symmedians as Hyperbolic Barycenters |
| title_fullStr | Symmedians as Hyperbolic Barycenters |
| title_full_unstemmed | Symmedians as Hyperbolic Barycenters |
| title_short | Symmedians as Hyperbolic Barycenters |
| title_sort | symmedians as hyperbolic barycenters |
| topic | symmedian point Hyperbolic barycenter harmonic polygon ideal hyperbolic polygon |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.677/ |
| work_keys_str_mv | AT arnoldmaxim symmediansashyperbolicbarycenters AT arrechecarlose symmediansashyperbolicbarycenters |