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Exact Computation of the Topology and Geometric Invariants of the Voronoi Diagram of Spheres in 3D

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Abstract

In this paper, we are addressing the exact computation of the Delaunay graph (or quasi-triangulation) and the Voronoi diagram of spheres using Wu’s algorithm. Our main contributions are first a methodology for automated derivation of invariants of the Delaunay empty circumsphere predicate for spheres and the Voronoi vertex of four spheres, then the application of this methodology to get all geometrical invariants that intervene in this problem and the exact computation of the Delaunay graph and the Voronoi diagram of spheres. To the best of our knowledge, there does not exist a comprehensive treatment of the exact computation with geometrical invariants of the Delaunay graph and the Voronoi diagram of spheres. Starting from the system of equations defining the zero-dimensional algebraic set of the problem, we are applying Wu’s algorithm to transform the initial system into an equivalent Wu characteristic (triangular) set. In the corresponding system of algebraic equations, in each polynomial (except the first one), the variable with higher order from the preceding polynomial has been eliminated (by pseudo-remainder computations) and the last polynomial we obtain is a polynomial of a single variable. By regrouping all the formal coefficients for each monomial in each polynomial, we get polynomials that are invariants for the given problem. We rewrite the original system by replacing the invariant polynomials by new formal coefficients. We repeat the process until all the algebraic relationships (syzygies) between the invariants have been found by applying Wu's algorithm on the invariants. Finally, we present an incremental algorithm for the construction of Voronoi diagrams and Delaunay graphs of spheres in 3D and its application to Geodesy.

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References

  1. [1]

    Green P, Sibson R (1978) Computing dirichlet tessellations in the plane. The Computer Journal 21(2):168–173

  2. [2]

    Brown KQ (1979) Voronoi diagrams from convex hulls. Information Processing Letters 9(5):223–228

  3. [3]

    Aurenhammer F. Gewichtete Voronoi diagramme: Geometrische deutung und Konstruktions-Algorithmen [Ph.D. Thesis]. IIG-TU Graz, Austria, 1984.

  4. [4]

    Aurenhammer F, Klein R. Voronoi diagrams. In Handbook of Computational Geometry, Chapter V, Sack J, Urrutia G (eds.), Elsevier Science Publishing, 2000, pp.201–290.

  5. [5]

    Guibas L J, Knuth D E, Sharir M. Randomized incremental construction of Delaunay and Voronoi diagrams. In Proc. the 17th International Colloquium on Automata, Languages and Programming, July 1990, pp.414–431.

  6. [6]

    Okabe A, Boots B, Sugihara K. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. John Wiley and Sons, 1992.

  7. [7]

    Kim DS, Cho Y, Kim D (2005) Euclidean Voronoi diagram of 3D balls and its computation via tracing edges. Computer-Aided Design 37(13):1412–1424

  8. [8]

    Kim D, Kim DS (2006) Region-expansion for the Voronoi diagram of 3D spheres. Comput Aided Des 38(5):417–430

  9. [9]

    Ryu J, Kim D, Cho Y, Park R, Kim D S. Computation of molecular surface using Euclidean Voronoi diagram. Computer-Aided Design and Applications, 2005, 2(1/4).

  10. [10]

    Kim DS, Cho Y, Kim D, Kim S, Bhak J, Lee SH (2005) Euclidean Voronoi diagrams of 3D spheres and applications to protein structure analysis. Japan Journal of Industrial and Applied Mathematics 22(2):251–265

  11. [11]

    Will H M. Fast and efficient computation of additively weighted Voronoi cells for applications in molecular biology. In Proc. the 6th Scandinavian Workshop on Algorithm Theory, July 1998, pp.310–321.

  12. [12]

    Gavrilova M. Proximity and applications in general metrics [Ph.D. Thesis]. University of Calgary, Alberta, Canada, 1998.

  13. [13]

    Gavrilova M, Rokne J (2003) Updating the topology of the dynamic Voronoi diagram for spheres in Euclidean d-dimensional space. Comput Aided Geom Des 20(4):231–242

  14. [14]

    Gavrilova M (2009) An explicit solution for computing the vertices of the Euclidean d-dimensional Voronoi diagram of spheres in a floating-point arithmetic. International Journal of Computational Geometry and Applications 19(5):415–424

  15. [15]

    Wu WT, Gao XS (2006) Automated reasoning and equation solving with the characteristic set method. J Comput Sci & Technol 21(5):756–764

  16. [16]

    Nishida T, Sugihara K. Precision necessary for d-dimensional sphere Voronoi diagrams. In Proc. the 5th International Symposium on Voronoi Diagrams in Science and Engineering, September 2008, pp.157–167.

  17. [17]

    Nishida T, Tanaka Y, Sugihara K. Evaluation of the precision for exact computation of a circle Voronoi diagram. Technical Report UW-CS-TR-1481, Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Japan, October 2007.

  18. [18]

    Kim D S, Cho Y, Kim D. Calculating three-dimensional (3D) Voronoi diagrams. Patent No. US7825927, 2010.

  19. [19]

    Hanniel I, Elber G (2009) Computing the Voronoi cells of planes, spheres and cylinders in R3. Comput Aided Geom Des 26(6):695–710

  20. [20]

    Anton F. Voronoi diagrams of semi-algebraic sets [Ph.D. Thesis]. The University of British Columbia, Vancouver, Canada, 2004.

  21. [21]

    Anton F. A certified Delaunay graph conflict locator for semi-algebraic sets. In Proc. Int. Conf. Computational Science and Its Applications, Part I, May 2005, pp.669–682.

  22. [22]

    Kim DS, Kim D, Cho Y, Sugihara K (2006) Quasi-triangulation and interworld data structure in three dimensions. Computeraided Design 38(7):808–819

  23. [23]

    Kim DS, Cho Y, Sugihara K (2010) Quasi-worlds and quasi-operators on quasi-triangulations. Comput Aided Des 42(10):874–888

  24. [24]

    Anton F, Mioc D, Gold C. The Voronoi diagram of circles and its application to the visualization of the growth of particles. In Transactions on Computational Science III, Gavrilova M, Tan C J (eds.), Berlin, Heidelberg: Springer, 2009, pp.20–54.

  25. [25]

    Kolmogorov A. A statistical theory for the recrystallization of metals. Akad. nauk SSSR, Izv., Ser. Matem., 1937, (3): 355–359.

  26. [26]

    Deschamps A. Handbook of Aluminum, New York, USA: Marcel Dekker, Inc., 2005, pp.155–192.

  27. [27]

    Chen Z, Xu J. Robust algorithm for k-gon Voronoi diagram construction. In Proc. the 14th Canadian Conference on Computational Geometry, August 2002, pp.77–81.

  28. [28]

    Blum L, Cucker F, Shub M, Smale S. Complexity and Real Computation. New York: Springer-Verlag, 1997.

  29. [29]

    Canny JF, Emiris IZ (2000) A subdivision-based algorithm for the sparse resultant. J ACM 47(3):417–451

  30. [30]

    Voronoï GF (1908) Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier mémoire. sur quelques propriétés des formes quadratiques positives parfaites. Journal für die reine und angewandte Mathematik 1908(133):97–178

  31. [31]

    Voronoï G F. Nouvelles applications des paramμetres continus à la théorie des formes quadratiques. deuxiμeme mémoire. recherches sur les paralléloμedres primitifs. premiμere partie. partition uniforme de l’espace analytique à n dimensions à l’aide des translations d’un même polyμedre convexe. Journal für die reine und angewandte Mathematik, 1908, 1908(134): 198–287. (In French)

  32. [32]

    Voronoï G F. Nouvelles applications des paramμetres continus à la théorie des formes quadratiques. deuxième mémoire. recherches sur les paralléloèdres primitifs. seconde partie. domaines de formes quadratiques correspondant aux différents types de paralléloèdres primitifs. Journal für die reine und angewandte Mathematik, 1909, 1909(136): 67–182. (In French)

  33. [33]

    Greuel G M, Pfister G. A Singular Introduction to Commutative Algebra. Berlin: Springer-Verlag, 2002.

  34. [34]

    Hamelryck T (2005) An amino acid has two sides: A new 2D measure provides a different view of solvent exposure. Proteins: Structure, Function, and Bioinformatics 59(1):38–48

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Correspondence to François Anton.

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Anton, F., Mioc, D. & Santos, M. Exact Computation of the Topology and Geometric Invariants of the Voronoi Diagram of Spheres in 3D. J. Comput. Sci. Technol. 28, 255–266 (2013). https://doi.org/10.1007/s11390-013-1327-3

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Keywords

  • Voronoi diagram of spheres
  • Delaunay graph of spheres
  • Wu's method
  • invariant
  • characteristic set