International Journal of CAD/CAM

Society for Computational Design and Engineering (한국CDE학회)
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Topology Representation for the Voronoi Diagram of 3D Spheres

  • Cho, Young-Song (Voronoi Diagram Research Center, Hanyang University) ;
  • Kim, Dong-Uk (Voronoi Diagram Research Center, Hanyang University) ;
  • Kim, Deok-Soo (Department of Industrial Engineering, Hanyang University)
  • Published : 2005.12.01
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Abstract

Euclidean Voronoi diagram of spheres in 3-dimensional space has not been explored as much as it deserves even though it has significant potential impacts on diverse applications in both science and engineering. In addition, studies on the data structure for its topology have not been reported yet. Presented in this, paper is the topological representation for Euclidean Voronoi diagram of spheres which is a typical non-manifold model. The proposed representation is a variation of radial edge data structure capable of dealing with the topological characteristics of Euclidean Voronoi diagram of spheres distinguished from those of a general non-manifold model and Euclidean Voronoi diagram of points. Various topological queries for the spatial reasoning on the representation are also presented as a sequence of adjacency relationships among topological entities. The time and storage complexities of the proposed representation are analyzed.

Keywords

References

  1. Angelov, B., Sadoc, J.-F., Jullien, R., Soyer, A., Mornon J.-P., and Chomilier, J. (2002), Nonatomic solvent-driven Voronoi tessellation of proteins: an open tool to analyze protein folds, Proteins: Structure, Function, and Genetics, 49(4), 446-456 https://doi.org/10.1002/prot.10220
  2. Aurenhammer, F. (1987), Power diagrams: properties, algorithms and applications, SIAM Journal of Computing, 16, 78-96 https://doi.org/10.1137/0216006
  3. Bajaj, C.L., Pascucci, V., Shamir, A., Holt, R.J., and Netravali, A.N. (2003), Dynamic maintenance and visualization of molecular surfaces, Discrete Applied Mathematics, 127, 23-51 https://doi.org/10.1016/S0166-218X(02)00283-4
  4. Boissonnat, J.D. and Karavelas, M.l. (2003), On the combinatorial complexity of Euclidean Voronoi cells and convex hulls of d-dimensional spheres, Proceedings of the 14th annual ACM-SIAM Symposium on Discrete Algorithms, 305-312
  5. Brisson, E. (1989) Representing geometric structures in d dimensions: topology and order, Proceedings of 5th Symposium on Computational Geometry, ACM Press, New York, 218-227 https://doi.org/10.1145/73833.73858
  6. Choi, Y. (1989), Vertex-based Boundary Representation of Non-manifold Geometric Models, Ph.D. Dissertation, Carnegie Mellon University, USA
  7. Connolly, M.L. (1983), Analytical molecular surface calculation, Journal of Applied Crystallography, 16, 548-558 https://doi.org/10.1107/S0021889883010985
  8. Connolly, M.L. (1983), Solvent-accessible surfaces of proteins and nucleic acids, Science, 221, 709-713 https://doi.org/10.1126/science.6879170
  9. Connolly, M.L. (1996), Molecular surfaces: a review. Network Sci
  10. Dafas, P., Bolser, D., Gomoluch, J., Park, J., and Schroeder, M. (2004), Using convex hulls to compute protein interactions from known structures, Bioinformatics, 20(10),1486-1490 https://doi.org/10.1093/bioinformatics/bth106
  11. Dobkin, D.P. and Laszlo, M.J. (1987), Primitives for the Manipulation of Three-Dimensional Subdivisions, Proceedings of 3rd Symposium on Computational Geometry, ACM Press, New York, 86-99 https://doi.org/10.1145/41958.41967
  12. Edelsbrunner, H., Facello, M., and Liang, J. (1998), On the definition and the construction of pockets in macromolecules, Discrete Applied Mathematics, 88, 83-102 https://doi.org/10.1016/S0166-218X(98)00067-5
  13. Gavrilova, M. (1998), Proximity and Applications in General Metrics, Ph. D. Thesis, The University of Calgary, Dept. of Computer Science, Calgary, AB, Canada
  14. Gavrilova, M. and Rokne, J. (2003), Updating the topology of the dynamic Voronoi diagram for spheres in Euclidean d-dimensional space, Computer Aided Geometric Design, 20(4), 231-242 https://doi.org/10.1016/S0167-8396(03)00027-X
  15. Goede, A., Preissner, R., and Frommel, C. (1997), Voronoi cell: new method for allocation of space among atoms: elimination of avoidable errors in calculation of atomic volume and density, Journal of Computational Chemistry, 18(9), 1113-1123 https://doi.org/10.1002/(SICI)1096-987X(19970715)18:9<1113::AID-JCC1>3.0.CO;2-U
  16. Kim, D.-S., Chung, Y.-C., Kim, J.J., Kim, D., and Yu, K. (2002), Voronoi diagram as an analysis tool for spatial properties for ceramics, Journal of Ceramic Processing, Research, 3(3), 150-152
  17. Kim, D.-S., Kim, D., and Sugihara, K. (2001), Voronoi diagram of a circle set from Voronoi diagram of a point set: I. Topology, Computer Aided Geometric Design, 18(6), 541-562 https://doi.org/10.1016/S0167-8396(01)00050-4
  18. Kim, D.-S., Kim, D., and Sugihara, K. (2001), Voronoi diagram of a circle set from Voronoi diagram of a point set: II. Geometry, Computer Aided Geometric Design 18(6), 563-585 https://doi.org/10.1016/S0167-8396(01)00051-6
  19. Kim, D.-S., Cho, Y., Kim, D., and Cho, C.-H. (2004), Protein structure analysis using Euclidean Voronoi diagram of atoms, Proceedings of International Workshop on Biometric Technologies (BT 2004), Special Forum on Modeling and Simulation in Biometric Technology, 125-129
  20. Kim, D.-S., Cho, Y., and Kim, D. (2005), Euclidean Voronoi diagram of 3D balls and its computation via tracing edges, Computer-Aided Design 37(13), 1412-1424 https://doi.org/10.1016/j.cad.2005.02.013
  21. Lee, K. (1999), Principles of CAD/CAM/CAE Systems, Addison Wesley
  22. Lee, S.H. and Lee, K. (2001), Partial entity structure: a compact boundary representation for non-manifold geometric modeling, ASME Journal of Computing & Information Science in Engineering, 1(4), 356-365 https://doi.org/10.1115/1.1433486
  23. Lienhardt, P. (1989), Subdivisions of n-dimensional spaces and n-dimensional generalized map, Proceedings of 5th Symposium on Computational Geometry, ACM Press, New York, 228-236 https://doi.org/10.1145/73833.73859
  24. Luchnikov, V.A., Medvedev, N.N., Oger, L., and Troadec, J.-P. (1999), Voronoi-Delaunay analysis of voids in systems of nonspherical particles. Physical Review E, 59(6), 7205-7212 https://doi.org/10.1103/PhysRevE.59.7205
  25. Montoro, J.C.G. and Abascal, J.L.F. (1993), The Voronoi polyhedra as tools for structure determination in simple disordered systems, The Journal of Physical Chemistry, 97(16), 4211-4215 https://doi.org/10.1021/j100118a044
  26. Ohno, K., Esfarjani, K., and Kawazoe, Y. (1999), Computational Materials Science From Ab Initio To Monte Carlo Methods, Springer
  27. Okabe, A., Boots, B., and Sugihara, K. (1992), Spatial Tessellations: Concepts and Applications of Voronoi, Diagram, John Wiley & Sons
  28. Peters, K.P., Fauck, J., and Frommel, C. (1996), The automatic search for ligand binding sites in protein of know three-dimensional structure using only geometric criteria, Journal of Molecular Biology, 256, 201-213 https://doi.org/10.1006/jmbi.1996.0077
  29. RCSB Protein Data Bank (April 8, 2004), http://www.rcsb.org/pdb/
  30. Richards, F.M. (1974), The interpretation of protein structures: total volume, group volume distributions and packing density, Journal of Molecular Biology, 82, 1-14 https://doi.org/10.1016/0022-2836(74)90570-1
  31. Richards, F.M. (1977), Areas, volumes, packing and protein structure, Annu. Rev. Biophys. Bioeng, 6, 151-176 https://doi.org/10.1146/annurev.bb.06.060177.001055
  32. Voloshin, V.P., Beaufils, S., and Medvedev, N.N. (2002), Void space analysis of the structure of liquids, Journal of Molecular Liquids, 96-97, 101-112 https://doi.org/10.1016/S0167-7322(01)00330-0
  33. Weiler, K. (1988), The radial edge structure: a topological representation for non-manifold geometric boundary modeling, in: Wozny, M.J., McLaughlin, H.W. and Encarnacao, J.L. Eds., Geometric Modeling for CAD Applications, North Holland, Elsevier Science Publishers B.V., 3-36
  34. Will, H.-M. (1999), Computation of Additively Weighted Voronoi Cells for Applications in Molecular Biology, Ph.D. Dissertation, Swiss Federal Institute of Technology, Zurich
  35. Yamaguchi, Y. and Kimura, F. (1995), Nonmanifold topology based on coupling entities, IEEE Computer Graphics and Applications, 15(1), 42-50 https://doi.org/10.1109/38.364963