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Polyline‐sourced Geodesic Voronoi Diagrams on Triangle Meshes
Author(s) -
Xu Chunxu,
Liu YongJin,
Sun Qian,
Li Jinyan,
He Ying
Publication year - 2014
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.12484
Subject(s) - voronoi diagram , geodesic , centroidal voronoi tessellation , power diagram , weighted voronoi diagram , polygon mesh , computer science , computational geometry , mathematics , algorithm , triangle mesh , euclidean distance , mesh generation , euclidean geometry , diagram , topology (electrical circuits) , combinatorics , geometry , physics , database , finite element method , thermodynamics
This paper studies the Voronoi diagrams on 2‐manifold meshes based on geodesic metric (a.k.a. geodesic Voronoi diagrams or GVDs), which have polyline generators. We show that our general setting leads to situations more complicated than conventional 2D Euclidean Voronoi diagrams as well as point‐source based GVDs, since a typical bisector contains line segments, hyperbolic segments and parabolic segments. To tackle this challenge, we introduce a new concept, called local Voronoi diagram (LVD), which is a combination of additively weighted Voronoi diagram and line‐segment Voronoi diagram on a mesh triangle. We show that when restricting on a single mesh triangle, the GVD is a subset of the LVD and only two types of mesh triangles can contain GVD edges. Based on these results, we propose an efficient algorithm for constructing the GVD with polyline generators. Our algorithm runs in O(nNlogN) time and takes O(nN) space on an n‐face mesh with m generators, where N = max {m, n}. Computational results on real‐world models demonstrate the efficiency of our algorithm.