Properties for Voronoi diagrams of arbitrary order in the sphere
Document typeMaster thesis
Rights accessOpen Access
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.
DegreeMÀSTER UNIVERSITARI EN MATEMÀTICA AVANÇADA I ENGINYERIA MATEMÀTICA (Pla 2010)