Journal of Computational Design and Engineering

Society for Computational Design and Engineering (한국CDE학회)
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Voronoi diagrams, quasi-triangulations, and beta-complexes for disks in R2: the theory and implementation in BetaConcept

  • Kim, Jae-Kwan (Voronoi Diagram Research Center, Hanyang University) ;
  • Cho, Youngsong (Voronoi Diagram Research Center, Hanyang University) ;
  • Kim, Donguk (Department of Industrial, Information, and Management Engineering, Gangneung-Wonju National University) ;
  • Kim, Deok-Soo (Department of Mechanical Engineering, Hanyang University)
  • Received : 2013.09.17
  • Accepted : 2013.11.01
  • Published : 2014.04.01
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Abstract

Voronoi diagrams are powerful for solving spatial problems among particles and have been used in many disciplines of science and engineering. In particular, the Voronoi diagram of three-dimensional spheres, also called the additively-weighted Voronoi diagram, has proven its powerful capabilities for solving the spatial reasoning problems for the arrangement of atoms in both molecular biology and material sciences. In order to solve application problems, the dual structure, called the quasi-triangulation, and its derivative structure, called the beta-complex, are frequently used with the Voronoi diagram itself. However, the Voronoi diagram, the quasi-triangulation, and the beta-complexes are sometimes regarded as somewhat difficult for ordinary users to understand. This paper presents the two-dimensional counterparts of their definitions and introduce the BetaConcept program which implements the theory so that users can easily learn the powerful concept and capabilities of these constructs in a plane. The BetaConcept program was implemented in the standard C++ language with MFC and OpenGL and freely available at Voronoi Diagram Research Center (http://voronoi.hanyang.ac.kr).

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References

  1. Okabe A, Boots B, Sugihara K, Chiu SN. Spatial tessellations: Concepts and applications of Voronoi diagrams. 2nd ed. Chichester: John Wiley & Sons; 2000. 696 p.
  2. Cho Y, Kim JK, Ryu J, Won CI, Kim CM, Kim D, Kim DS. BetaMol: A molecular modeling, analysis and visualization software based on the beta-complex and the quasitriangulation. Journal of Advanced Mechanical Design, Systems, and Manufacturing. 2012; 6(3): 389-403. https://doi.org/10.1299/jamdsm.6.389
  3. Cho Y, Kim D, Kim DS. Topology representation for the Voronoi diagram of 3D spheres. International Journal of CAD/CAM. 2005; 5(1): 59-68.
  4. Kim DS, Cho Y, Kim D. Euclidean Voronoi diagram of 3D balls and its computation via tracing edges. Computer-Aided Design. 2005; 37(13): 1412-1424. https://doi.org/10.1016/j.cad.2005.02.013
  5. Kim D, Kim DS. Region-expansion for the Voronoi diagram of 3D spheres. Computer-Aided Design. 2006; 38(5): 417-430. https://doi.org/10.1016/j.cad.2005.11.007
  6. Lee K. Principles of CAD/CAM/CAE systems. Upper Saddle River (NJ): Prentice Hall; 1999. 640 p.
  7. Lee D, Drysdale R. Generalization of Voronoi diagrams in the plane. SIAM Journal on Computing. 1981; 10(1): 73-87. https://doi.org/10.1137/0210006
  8. Sharir M. Intersection and closest-pair problems for a set of planar discs. SIAM Journal on Computing. 1985; 14(2): 448-468. https://doi.org/10.1137/0214034
  9. Yap CK. An O(n log n) algorithm for the Voronoi diagram of a set of simple curve segments. Discrete & Computational Geometry. 1987; 2: 365-393. https://doi.org/10.1007/BF02187890
  10. Gavrilova M, Rokne J. Swap conditions for dynamic vd for circles and line segments. Computer Aided Geometric Design. 1999; 16(2): 89-106. https://doi.org/10.1016/S0167-8396(98)00039-9
  11. Sugihara K. Approximation of generalized Voronoi diagrams by ordinary Voronoi diagrams. Graphical Models and Image Processing. 1993; 55(6): 522-531. https://doi.org/10.1006/cgip.1993.1039
  12. Kim DS, Kim D, Sugihara K. Voronoi diagram of a circle set from Voronoi diagram of a point set: I. topology. Computer Aided Geometric Design. 2001; 18: 541-562. https://doi.org/10.1016/S0167-8396(01)00050-4
  13. Kim DS, Kim D, Sugihara K. Voronoi diagram of a circle set from Voronoi diagram of a point set: II. geometry. Computer Aided Geometric Design. 2001; 18: 563-585. https://doi.org/10.1016/S0167-8396(01)00051-6
  14. Kim DS, Kim D, Cho Y, Sugihara K. Quasi-triangulation and interworld data structure in three dimensions. Computer-Aided Design. 2006; 38(7): 808-819. https://doi.org/10.1016/j.cad.2006.04.008
  15. Kim DS, Cho Y, Sugihara K. Quasi-worlds and quasioperators on quasi-triangulations. Computer-Aided Design. 2010; 42(10): 874-888. https://doi.org/10.1016/j.cad.2010.06.002
  16. Kim DS, Cho Y, Ryu J, Kim JK, Kim D. Anomalies in quasitriangulations and beta-complexes of spherical atoms in molecules. Computer-Aided Design. 2013; 45(1): 35-52. https://doi.org/10.1016/j.cad.2012.03.005
  17. Kim DS, Cho Y, Sugihara K, Ryu J, Kim D. Threedimensional beta-shapes and beta-complexes via quasitriangulation. Computer-Aided Design. 2010; 42(10): 911-929. https://doi.org/10.1016/j.cad.2010.06.004
  18. Kim DS, Kim JK, Cho Y, Kim CM. Querying simplexes in quasi-triangulation. Computer-Aided Design. 2012; 44(2): 85-98. https://doi.org/10.1016/j.cad.2011.09.010
  19. Kim DS. Polygon offsetting using a Voronoi diagram and two stacks. Computer-Aided Design. 1998; 30(14): 1069-1076. https://doi.org/10.1016/S0010-4485(98)00063-3

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