Euclidean Voronoi Diagrams of 3D Spheres: Their Construction and Related Problems from Biochemistry

  • Deok-Soo Kim
  • Donguk Kim
  • Youngsong Cho
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3604)


Voronoi diagrams have several important applications in science and engineering. While the properties and algorithms for the ordinary Voronoi diagrams of point sets have been well-known, their counterparts for a set of spheres have not been sufficiently studied.

In this paper, we present properties and two algorithms for Voronoi diagrams of 3D spheres based on the Euclidean distance from the surface of spheres. Starting from a valid initial Voronoi vertex, the edge-tracing algorithm follows Voronoi edges until the construction is completed. The region-expansion algorithm constructs the desired diagram by successively expanding the Voronoi region of each sphere, one after another, via a series of topology operations, starting from the ordinary Voronoi diagram for the centres of spheres.

In the worst-case, the edge-tracing algorithm takes O(mn) time, and the region-expansion algorithm takes O(n 3 log n) time, where m and n are the numbers of edges and spheres, respectively. It should, however, be noted that the worst-case time complexity for both algorithms reduce to O(n 2) for proteins since the number of immediate neighbor atoms for an atom is constant. Adapting appropriate filtering techniques to reduce search space, the expected time complexities can even reduce to linear.

Then, we show how such a Voronoi diagram can be used for solving various important geometric problems in biological systems by illustrating two examples: the computation of surfaces defined on a protein, and the extraction and characterization of interaction interfaces between multiple proteins.


Voronoi Diagram Molecular Surface Voronoi Region Tangent Sphere Power Diagram 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Angelov, B., Sadoc, J.-F., Jullien, R., Soyer, A., Mornon, J.-P., Chomilier, J.: Nonatomic solvent-driven Voronoi tessellation of proteins: an open tool to analyze protein folds. Proteins: Structure, Function, and Genetics 49(4), 446–456 (2002)CrossRefGoogle Scholar
  2. 2.
    Aurenhammer, F.: Power diagrams: properties, algorithms and applications. SIAM Journal of Computing 16, 78–96 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bajaj, C.L., Pascucci, V., Shamir, A., Holt, R.J., Netravali, A.N.: Dynamic maintenance and visualization of molecular surfaces. Discrete Applied Mathematics 127, 23–51 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Blum, H.: A transformation for extracting new descriptors of shape. In: Proc. Symp. Models for Perception of Speech & Visual Form, pp. 362–380 (1967)Google Scholar
  5. 5.
    Blum, H., Nagel, R.N.: Shape description using weighted symmetric axis features. Pattern Recognition 10, 167–180 (1978)zbMATHCrossRefGoogle Scholar
  6. 6.
    Boissonnat, J.D., Karavelas, M.I.: On the combinatorial complexity of Euclidean Voronoi cells and convex hulls of d-dimensional spheres. In: Proceedings of the 14th annual ACM-SIAM Symposium on Discrete Algorithms, pp. 305–312 (2003)Google Scholar
  7. 7.
    Cho, Y., Kim, D., Kim, D.-S.: Topology representation for Euclidean Voronoi diagram of spheres in 3D. In: Proc. Digital Engineering Workshop/5th Japan-Korea CAD/CAM Workshop (2005)Google Scholar
  8. 8.
    Connolly, M.L.: Analytical molecular surface calculation. Journal of Applied Crystallography 16, 548–558 (1983)CrossRefGoogle Scholar
  9. 9.
    Connolly, M.L.: Solvent-accessible surfaces of proteins and nucleic acids. Science 221, 709–713 (1983)CrossRefGoogle Scholar
  10. 10.
    Connolly, M.L.: Molecular surfaces: a review. Network Sci. (1996)Google Scholar
  11. 11.
    Edelsbrunner, H., MÜcke, E.P.: Three-dimensional alpha shapes. ACM Transactions on Graphics 13(1), 43–72 (1994)zbMATHCrossRefGoogle Scholar
  12. 12.
    Edelsbrunner, H., Facello, M., Liang, J.: On the definition and the construction of pockets in macromolecules. Discrete Applied Mathematics 88, 83–102 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gavrilova, M.: Proximity and Applications in General Metrics. Ph.D. thesis: The University of Calgary, Dept. of Computer Science, Calgary, AB, Canada (1998)Google Scholar
  14. 14.
    Gavrilova, M., Rokne, J.: Updating the topology of the dynamic Voronoi diagram for spheres in Euclidean d-dimensional space. Computer Aided Geometric Design 20(4), 231–242 (2003)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Goede, A., Preissner, R., Frömmel, C.: 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 (1997)CrossRefGoogle Scholar
  16. 16.
    Halperin, D., Overmars, M.H.: Spheres, Molecules, and Hidden Surface Removal. In: Proc. 10th ACM Symposium on Computational Geometry, pp. 113–122 (1994)Google Scholar
  17. 17.
    Held, M.: On the Computational Geometry of Pocket Machining. LNCS, vol. 500. Springer, Heidelberg (1991)zbMATHGoogle Scholar
  18. 18.
    Kim, D.-S., Kim, D., Sugihara, K.: Voronoi diagram of a circle set from Voronoi diagram of a point set: I. Topology. Computer Aided Geometric Design 18(6), 541–562 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kim, D.-S., Kim, D., Sugihara, K.: Voronoi diagram of a circle set from Voronoi diagram of a point set: II. Geometry. Computer Aided Geometric Design 18(6), 563–585 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kim, D.-S., Cho, Y., Kim, D., Cho, C.-H.: Protein structure analysis using Euclidean Voronoi diagram of atoms. In: Proc. International Workshop on Biometric Technologies (BT 2004), pp. 125–129 (2004)Google Scholar
  21. 21.
    Kim, D.-S., Cho, Y., Kim, D.: Edge-tracing algorithm for Euclidean Voronoi diagram of 3D spheres. In: Proc. 16th Canadian Conference on Computational Geometry, pp. 176–179 (2004)Google Scholar
  22. 22.
    Kim, D.S., Cho, S., Kim, Y., Kim, D., Bhak, J.: Euclidean Voronoi Diagram of 3D Spheres and Applications to Protein Structure Analysis. In: International Symposium on Voronoi Diagrams in Science and Engineering, pp. 13–15. University of Tokyo, Tokyo (2004)Google Scholar
  23. 23.
    Kim, D.-S., Cho, Y., Kim, D.: Euclidean Voronoi diagram of 3D balls and its computation via tracing edges. Computer-Aided Design (in printing)Google Scholar
  24. 24.
    Kim, D., Kim, D.-S.: Euclidean Voronoi diagrams for spheres in 3D by expanding regions (in prepration)Google Scholar
  25. 25.
    Kirkpatrick, D.G.: Efficient computation of continuous skeletons. In: Proc. 14th IEEE Symp. Foundations of Computer Science, pp. 18–27 (1979)Google Scholar
  26. 26.
    Lee, B., Richards, F.M.: The interpretation of protein structures: estimation of static accessibility. Journal of Molecular Biology 55, 379–400 (1971)CrossRefGoogle Scholar
  27. 27.
    Lee, D.T.: Medial axis transformation of a planar shape. IEEE Transactions on Pattern Analysis and Machine Intelligence 4, 363–369 (1982)zbMATHCrossRefGoogle Scholar
  28. 28.
    Law, A.M., Kelton, W.D.: Simulation Modeling and Analysis. McGraw-Hill, New York (1982)zbMATHGoogle Scholar
  29. 29.
    Luchnikov, V.A., Medvedev, N.N., Oger, L., Troadec, J.-P.: Voronoi-Delaunay analyzis of voids in systems of nonspherical particles. Physical review E 59(6), 7205–7212 (1999)CrossRefGoogle Scholar
  30. 30.
    Montoro, J.C.G., Abascal, J.L.F.: The Voronoi polyhedra as tools for structure determination in simple disordered systems. The Journal of Physical Chemistry 97(16), 4211–4215 (1993)CrossRefGoogle Scholar
  31. 31.
    Naberukhin, Y.I., Voloshin, V.P., Medvedev, N.N.: Geometrical analysis of the structure of simple liquids: percolation approach. Molecular Physics 73, 917–936 (1991)CrossRefGoogle Scholar
  32. 32.
    Noggle, J.H.: Physical Chemistry, 3rd edn. Freedom Academy Publishing Co. (1996)Google Scholar
  33. 33.
    Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. John Wiley & Sons, Chichester (1999)Google Scholar
  34. 34.
    Peters, K.P., Fauck, J., Frömmel, C.: The automatic search for ligand binding sites in protein of know three-dimensional strucutre using only geometric criteria. Journal of Molecular Biology 256, 201–213 (1996)CrossRefGoogle Scholar
  35. 35.
    RCSB Protein Data Bank (2004),
  36. 36.
    Richards, F.M.: The interpretation of protein structures: total volume, group volume distributions and packing density. Journal of Molecular Biology 82, 1–14 (1974)CrossRefGoogle Scholar
  37. 37.
    Richards, F.M.: Areas, volumes, packing and protein structure. Annu. Rev. Biophys. Bioeng. 6, 151–176 (1977)CrossRefGoogle Scholar
  38. 38.
    Sastry, S., Corti, D.S., Debenedetti, P.G., Stillinger, F.H.: Statistical geometry of particle packings. I. Algorithm for exact determination of connectivity, volume, and surface areas of void space in monodisperse and polydisperse sphere packings. Physical Review E 56, 5524–5532 (1997)CrossRefMathSciNetGoogle Scholar
  39. 39.
    Thiessen, A.H.: Precipitation averages for large areas. Monthly Weather Review 39, 1082–1084 (1911)Google Scholar
  40. 40.
    Voloshin, V.P., Beaufils, S., Medvedev, N.N.: Void space analysis of the structure of liquids. Journal of Molecular Liquids 96-97, 101–112 (2002)CrossRefGoogle Scholar
  41. 41.
    Weiler, K.: The radial edge structure: a topological representation for non-manifold geometric boundary modeling. In: Wozny, M.J., McLaughlin, H.W., Encarnacao, J.L. (eds.) Geometric Modeling for CAD Applications, pp. 3–36. Elsevier Science Publishers, North Holland (1988)Google Scholar
  42. 42.
    Will, H.-M.: Computation of Additively Weighted Voronoi Cells for Applications in Molecular Biology. Ph.D. thesis, ETH, Zurich (1999)Google Scholar
  43. 43.
    Cambridge Crystallographic Data Centre (2005),

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Deok-Soo Kim
    • 1
  • Donguk Kim
    • 2
  • Youngsong Cho
    • 2
  1. 1.Department of Industrial EngineeringHanyang UniversitySeoulSouth Korea
  2. 2.Voronoi Diagram Research CenterHanyang UniversitySeoulSouth Korea

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