Polyline-sourced Geodesic Voronoi Diagrams on Triangle Meshes

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‐sou...

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Bibliographic Details
Published in:Computer graphics forum Vol. 33; no. 7; pp. 161 - 170
Main Authors: Xu, Chunxu, Liu, Yong-Jin, Sun, Qian, Li, Jinyan, He, Ying
Format: Journal Article
Language:English
Published: Oxford Blackwell Publishing Ltd 01.10.2014
Subjects:
Algorithms
Analysis
and object representations
Categories and Subject Descriptors (according to ACM CCS)
Computer graphics
I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Curve
I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve, surface, solid, and object representations
solid
Studies
surface
Topological manifolds
ISSN:0167-7055, 1467-8659
Online Access:Get full text
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Summary: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.
Bibliography:ark:/67375/WNG-18X0SHKM-S
Supporting Information
istex:BC38B30276BFD61D83AA16B93EF388F68E6C6B41
ArticleID:CGF12484
SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 14
ISSN:0167-7055
1467-8659
DOI:10.1111/cgf.12484