On the hyperbolicity of Delaunay triangulations
If $ X $ is a geodesic metric space and $ x_1, x_2, x_3\in X $, a <i>geodesic triangle</i> $ T = \{x_1, x_2, x_3\} $ is the union of the three geodesics $ [x_1 x_2] $, $ [x_2 x_3] $ and $ [x_3 x_1] $ in $ X $. The space $ X $ is <i>hyperbolic</i> if there exists a constant $...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2023-10-01
|
| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20231474?viewType=HTML |
| _version_ | 1797635376276307968 |
|---|---|
| author | Walter Carballosa José M. Rodríguez José M. Sigarreta |
| author_facet | Walter Carballosa José M. Rodríguez José M. Sigarreta |
| author_sort | Walter Carballosa |
| collection | DOAJ |
| description | If $ X $ is a geodesic metric space and $ x_1, x_2, x_3\in X $, a <i>geodesic triangle</i> $ T = \{x_1, x_2, x_3\} $ is the union of the three geodesics $ [x_1 x_2] $, $ [x_2 x_3] $ and $ [x_3 x_1] $ in $ X $. The space $ X $ is <i>hyperbolic</i> if there exists a constant $ \delta \ge 0 $ such that any side of any geodesic triangle in $ X $ is contained in the $ \delta $-neighborhood of the union of the two other sides. In this paper, we study the hyperbolicity of an important kind of Euclidean graphs called Delaunay triangulations. Furthermore, we characterize the Delaunay triangulations contained in the Euclidean plane that are hyperbolic. |
| first_indexed | 2024-03-11T12:20:14Z |
| format | Article |
| id | doaj.art-d9bdbf34da7449dba45c6c94874273d0 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-11T12:20:14Z |
| publishDate | 2023-10-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-d9bdbf34da7449dba45c6c94874273d02023-11-07T01:36:05ZengAIMS PressAIMS Mathematics2473-69882023-10-01812287802879010.3934/math.20231474On the hyperbolicity of Delaunay triangulationsWalter Carballosa0 José M. Rodríguez1José M. Sigarreta21. Department of Mathematics and Statistics, Florida International University, 11200 SW 8th Street, Miami, FL 33199, USA2. Departamento de Matemáticas, Universidad Carlos Ⅲ de Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain3. Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No.54 Col. Garita, 39650 Acalpulco Gro., MexicoIf $ X $ is a geodesic metric space and $ x_1, x_2, x_3\in X $, a <i>geodesic triangle</i> $ T = \{x_1, x_2, x_3\} $ is the union of the three geodesics $ [x_1 x_2] $, $ [x_2 x_3] $ and $ [x_3 x_1] $ in $ X $. The space $ X $ is <i>hyperbolic</i> if there exists a constant $ \delta \ge 0 $ such that any side of any geodesic triangle in $ X $ is contained in the $ \delta $-neighborhood of the union of the two other sides. In this paper, we study the hyperbolicity of an important kind of Euclidean graphs called Delaunay triangulations. Furthermore, we characterize the Delaunay triangulations contained in the Euclidean plane that are hyperbolic.https://www.aimspress.com/article/doi/10.3934/math.20231474?viewType=HTMLdelaunay triangulationvoronoi graphhyperbolic graphstessellation graphs |
| spellingShingle | Walter Carballosa José M. Rodríguez José M. Sigarreta On the hyperbolicity of Delaunay triangulations AIMS Mathematics delaunay triangulation voronoi graph hyperbolic graphs tessellation graphs |
| title | On the hyperbolicity of Delaunay triangulations |
| title_full | On the hyperbolicity of Delaunay triangulations |
| title_fullStr | On the hyperbolicity of Delaunay triangulations |
| title_full_unstemmed | On the hyperbolicity of Delaunay triangulations |
| title_short | On the hyperbolicity of Delaunay triangulations |
| title_sort | on the hyperbolicity of delaunay triangulations |
| topic | delaunay triangulation voronoi graph hyperbolic graphs tessellation graphs |
| url | https://www.aimspress.com/article/doi/10.3934/math.20231474?viewType=HTML |
| work_keys_str_mv | AT waltercarballosa onthehyperbolicityofdelaunaytriangulations AT josemrodriguez onthehyperbolicityofdelaunaytriangulations AT josemsigarreta onthehyperbolicityofdelaunaytriangulations |