Properties for Voronoi diagrams of arbitrary order in the sphere
In this thesis we study properties for spherical Voronoi diagrams of order $k$, SV_k(U)$ using different tools: the geometry of the sphere, a labeling for the edges of $SV_k(U)$, and the inversion transformation. Among the obtained properties, we show that $SV_k(U)$ has a small orientable cycle doub...
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| Tipo de recurso: | tesis de maestría |
| Fecha de publicación: | 2021 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/354664 |
| Acceso en línea: | https://hdl.handle.net/2117/354664 |
| Access Level: | acceso abierto |
| Palabra clave: | Discrete geometry Spherical Voronoi diagrams Higher order Voronoi diagram in the sphere Double cover Edge labeling Inversion transformation Alternating hexagon Geometria discreta Classificació AMS::52 Convex and discrete geometry::52C Discrete geometry Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria |
| Sumario: | In this thesis we study properties for spherical Voronoi diagrams of order $k$, SV_k(U)$ using different tools: the geometry of the sphere, a labeling for the edges of $SV_k(U)$, and the inversion transformation. Among the obtained properties, we show that $SV_k(U)$ has a small orientable cycle double cover, and we identify configurations that cannot appear in $SV_k(U)$ for small values of $k$. We generalize the construction of spherical Voronoi diagrams defined by Hyeon-Suk Na, Chung-Nim Lee and Otfried Cheong (2002) for order one to any order. We use that construction to prove that the numbers of faces, edges and vertices in $SV_k(U)$ are constant for fixed values of $k$ and $|U|$, i.e., do not depend on the positions of the points of $U$ on the sphere. Also, several connections and differences between Voronoi diagrams in the plane and in the sphere are addressed in this thesis. |
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