| Affiliation: | 1. School of Computer Science and Technology, Shandong University, Qingdao, China Peihui Wang and Na Yuan equally contribute to this paper;2. Ant Technology Group Co Ltd, China;3. School of Computer Science and Technology, Shandong University, Qingdao, China;4. School of Computer Science and Engineering, Nanyang Technological University, Singapore;5. School of Information and Technology, Qingdao University of Science and Technology, Qingdao, China;6. Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo, China;7. Department of Computer Science, The University of Hong Kong, China |
| Abstract: | 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. |
