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Robust Computation of 3D Apollonius Diagrams
Author(s) -
Wang Peihui,
Yuan Na,
Ma Yuewen,
Xin Shiqing,
He Ying,
Chen Shuangmin,
Xu Jian,
Wang Wenping
Publication year - 2020
Publication title -
computer graphics forum
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.578
H-Index - 120
eISSN - 1467-8659
pISSN - 0167-7055
DOI - 10.1111/cgf.14125
Subject(s) - voronoi diagram , power diagram , computer science , computation , vertex (graph theory) , centroidal voronoi tessellation , weighted voronoi diagram , euclidean geometry , robustness (evolution) , tracing , algorithm , polygon mesh , discretization , mathematics , graph , theoretical computer science , geometry , computer graphics (images) , mathematical analysis , biochemistry , chemistry , gene , operating system
Apollonius diagrams, also known as additively weighted Voronoi diagrams, are an extension of Voronoi diagrams, where the weighted distance is defined by the Euclidean distance minus the weight. The bisectors of Apollonius diagrams have a hyperbolic form, which is fundamentally different from traditional Voronoi diagrams and power diagrams. Though robust solvers are available for computing 2D Apollonius diagrams, there is no practical approach for the 3D counterpart. In this paper, we systematically analyze the structural features of 3D Apollonius diagrams, and then develop a fast algorithm for robustly computing Apollonius diagrams in 3D. Our algorithm consists of vertex location, edge tracing and face extraction, among which the key step is to adaptively subdivide the initial large box into a set of sufficiently small boxes such that each box contains at most one Apollonius vertex. Finally, we use centroidal Voronoi tessellation (CVT) to discretize the curved bisectors with well‐tessellated triangle meshes. We validate the effectiveness and robustness of our algorithm through extensive evaluation and experiments. We also demonstrate an application on computing centroidal Apollonius diagram.