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http://dx.doi.org/10.11627/jkise.2018.41.4.123

Triangulation of Voronoi Faces of Sphere Voronoi Diagram using Delaunay Refinement Algorithm  

Kim, Donguk (Department of Industrial and Management Engineering, Gangneung-Wonju National University)
Publication Information
Journal of Korean Society of Industrial and Systems Engineering / v.41, no.4, 2018 , pp. 123-130 More about this Journal
Abstract
Triangulation is one of the fundamental problems in computational geometry and computer graphics community, and it has huge application areas such as 3D printing, computer-aided engineering, surface reconstruction, surface visualization, and so on. The Delaunay refinement algorithm is a well-known method to generate quality triangular meshes when point cloud and/or constrained edges are given in two- or three-dimensional space. In this paper, we propose a simple but efficient algorithm to triangulate Voronoi surfaces of Voronoi diagram of spheres in 3-dimensional Euclidean space. The proposed algorithm is based on the Ruppert's Delaunay refinement algorithm, and we modified the algorithm to be applied to the triangulation of Voronoi surfaces in two ways. First, a new method to deciding the location of a newly added vertex on the surface in 3-dimensional space is proposed. Second, a new efficient but effective way of estimating approximation error between Voronoi surface and triangulation. Because the proposed algorithm generates a triangular mesh for Voronoi surfaces with guaranteed quality, users can control the level of quality of the resulting triangulation that their application problems require. We have implemented and tested the proposed algorithm for random non-intersecting spheres, and the experimental result shows the proposed algorithm produces quality triangulations on Voronoi surfaces satisfying the quality criterion.
Keywords
Voronoi Diagram; Voronoi Face; Delaunay Refinement Algorithm; Triangulation;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 Chew, L.P., Guaranteed-Quality Mesh Generation for Curved Surfaces, Proceedings of the 9th Annual Symposium on Computational Geometry, 1993, pp. 274-280.
2 Chew, L.P., Guaranteed-Quality Triangular Meshes, Technical Report TR-89-983, Cornell University, 1989.
3 de Berg, M., van Kreveld, M., Overmars, M., and Schwarzkopf, O., Computational Geometry : Algorithms and Applications. Springer, Berlin, 2nd edition, 2000.
4 Kim, C.-M., Won, C.-I., Cho, Y., Kim, D., Lee, S., Bhak, J., and Kim, D.-S., Interaction Interfaces in Proteins via the Voronoi Diagram of Atoms, Computer-Aided Design, 2006, Vol. 38, No. 11, pp. 1192-1204.   DOI
5 Kim, D. and Kim, D.-S., Region-Expansion for the Voronoi Diagram of 3D Spheres, Computer-Aided Design, 2006, Vol. 38, No. 5, pp. 417-430.   DOI
6 Kim, D., Acceleration of Delaunay Refinement Algorithm by Geometric Hashing, Korean Journal of Computational Design and Engineering, 2017, Vol. 22, No. 2, pp. 110-117.   DOI
7 Kim, D., Using Voronoi Diagram and Power Diagram in Application Problems, Journal of Society of Korea Industrial and Systems Engineering, 2012, Vol. 35, No. 4, pp. 235-243.   DOI
8 Kim, D.-S., Cho, Y., and Kim, D., Euclidean Voronoi Diagram of 3D Balls and Its Computation via Tracing Edges, Computer-Aided Design, 2005, Vol. 37, No. 13, pp. 1412-1424.   DOI
9 Lee, K., Principles of CAD/CAM/CAE Systems, Addison-Wesley, 1999.
10 Kim, D.-S., Kim, D., and Sugihara, K., Voronoi Diagram of a Circle Set from Voronoi Diagram of a Point Set : II. Geometry, Computer Aided Geometric Design, 2001, Vol. 18, pp. 563-585.   DOI
11 Okabe, A., Boots, B., Sugihara, K., and Chiu, S.N., Spatial Tessellations : Concepts and Applications of Voronoi Diagrams, 2nd edition, Chichester, John Wiley & Sons, 2000.
12 Ruppert, J., A Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh Generation, Journal of Algorithms, 1995, Vol. 18, No. 3, pp. 548-585.   DOI
13 Shewchuk, J.R., Delaunay Refinement Algorithms for Triangular Mesh Generation, Computational Geometry-Theory and Applications, 2002, Vol. 22, pp. 21-74.   DOI
14 Shewchuk, J.R., Lecture Notes on Delaunay Mesh Generation, Univ. of California at Berkeley, 2012.
15 Will, H.-M., Computation of Additively Weighted Voronoi Cells for Applications in Molecular Biology, Ph.D. dissertation, Swiss Federal Institute of Technology, Zurich, Switzerland, 1999.