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Computing the Topology of Voronoï Diagrams of Parallel Half-Lines

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Abstract

In this paper we consider the Voronoï diagram of a finite family of parallel half-lines, with the same orientation, constrained to a compact domain \({\mathscr {D}}_{0} \subset {\mathbb {R}}^3\), with respect to the Euclidean distance. We present an efficient approximation algorithm for computing such VD, using a subdivision process, which produces a mesh representing the topology of the VD in \({\mathscr {D}}_{0}\). The computed topology may not be correct for degenerate configurations or configurations close to degenerate. In this case, the output is a valid partition, which is close to the exact partition in Voronoï cells if the input data were given with no error. We also present the result of an implementation in Julia language with visualization using Axl software (axl.inria.fr) of the algorithm. Some examples and analysis are shown.

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References

  1. Adamou, I., Fioravanti, M., Gonzalez-Vega, L., Mourrain, B.: Bisectors and Voronoi diagram of a family of parallel half-lines. In: Dokken, T., Muntingh, G. (Eds.) SAGA—Advances in ShApes, Geometry, and Algebra, Geometry and Computing, pp. 241–279. Springer, Cham (2014)

  2. Alberti, L., Comte, G., Mourrain, B.: Meshing implicit algebraic surfaces: the smooth case. In: Schumaker, L.L., Maehlen, M., Morken, K. (Ed.) Mathematical Methods for Curves and Surfaces: Tromso’04, pp. 11–26. Nashboro (2005)

  3. Alberti, L., Mourrain, B.: Regularity criteria for the topology of algebraic curves and surfaces. In: Martin, R., Sabin, M., Winkler, J. (Eds.) Procedings of IMA international conference on Mathematics of Surfaces XII, pp. 1–28. Springer, Berlin (2007)

  4. Aurenhammer, F., Jüttler, B. Paulini, G.: Voronoi diagrams for parallel halflines and line segments in space. In: Okamoto, Y., Tokuyama,T. (Eds.) Proceedings of the 28th International Symposium Algorithms and Computing, vol. 92, pp. 7:1–7:10 (2017)

  5. Aurenhammer, F., Klein, R.: Voronoi diagrams. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 201–290. North-Holland, Amsterdam (2000)

    Chapter  Google Scholar 

  6. Aurenhammer, F., Paulini, G., Jüttler, B.: Voronoi diagrams for parallel halflines in 3D. In: Proceedings of the 32nd European Workshop on Computational Geometry, pp. 127–130 (2016)

  7. Barros, P.G., Pessoa, D.A., Leite, P.J.S., Teichrieb, V., Kelner, J.: Three dimensional oil well planning in ultra-deep water. In: Proceedings of Symposium on Virtual Reality (SVR), Belém, Parà, Vol. 1, pp. 285–296 (2006)

  8. Bentley, J.L.: Multidimensional binary search trees used for associative searching. Commun. ACM 18, 509–517 (1975)

    Article  Google Scholar 

  9. Boada, I., Coll, N., Madern, N., Antoni Sellares, J.: Approximations of 2d and 3d generalized Voronoi diagrams. Int. J. Comput. Math. 85(7), 1003–1022 (2008)

    Article  MathSciNet  Google Scholar 

  10. Boada, I., Coll, N., Sellares, J.: The Voronoi quadtree: construction and visualization. In: Eurographics 2002 Short Presentation, pp. 349–355 (2002)

  11. Boissonnat, J.-D., Wormser, C., Yvinec, M.: Curved Voronoi diagrams. In: Boissonnat, J.-D., Teillaud, M. (eds.) Effective Computational Geometry for Curves and Surfaces, pp. 67–116. Springer, Berlin (2006)

    Chapter  Google Scholar 

  12. Cao, T.-T., Tang, K., Mohamed, A., Tan, T.-S.: Parallel banding algorithm to compute exact distance transform with the GPU. In: Proceedings of the 2010 ACM SIGGRAPH symposium on Interactive 3D Graphics and Games, pp. 83–90. ACM (2010)

  13. Edelsbrunner, H., Seidel, R.: Voronoi diagrams and arrangements. Discrete Comput. Geom. 1, 25–44 (1946)

    Article  MathSciNet  Google Scholar 

  14. Edwards, J., Daniel, E., Pascucci, V., Bajaj, C.: Approximating the generalized Voronoi diagram of closely spaced objects. Comput. Graph. Forum 34(2), 299–309 (2015). https://doi.org/10.1111/cgf.12561

    Article  Google Scholar 

  15. Emiris, I., Malamatos, T., Tsigaridas, E.: Approximate nearest neighbor queries among parallel segments. In: Proceedings of the 26th European Workshop on Computational Geometry (2010)

  16. Emiris, I., Mantzaflaris, A., Mourrain, B.: Yet another algorithm for generalized Voronoï diagrams. In: Proceedings of the 27th Annual ACM Symposium on Applied Computing, pp. 109–110 (2012)

  17. Emiris, I.Z., Tsigaridas, E.P., Tzoumas, G.M.: Exact Delaunay graph of smooth convex pseudo-circles: general predicates, and implementation for ellipses. In: SIAM/ACM Joint Conference on Geometric and Physical Modeling, San Francisco, CA (2009)

  18. Etzion, M., Rappoport, A.: Computing Voronoi skeletons of a 3-d polyhedron by space subdivision. Comput. Geom. Theory Appl. 21(3), 87–120 (2002)

    Article  MathSciNet  Google Scholar 

  19. Everett, H., Gillot, C., Lazard, D., Lazard, S., Pouget, M.: The Voronoi diagram of three arbitrary lines \({\mathbb{R}}^3\). In: Abstracts of 25th European Workshop Computing Geometry (2009)

  20. Fan, C., Luo, J., Liu, J., Xu, Y.: Half-plane Voronoi Diagram. In: Proceedings of the 2011 Eighth International Symposium on Voronoi Diagrams in Science and Engineering, pp. 127–133

  21. Fortune, S.: Voronoi diagrams and Delaunay triangulations. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, pp. 377–388. CRC Press, Boca Raton (1997)

    MATH  Google Scholar 

  22. Fredrich, J.T., Coblentz, D., Fossum, A.F., Thorne, B.J.: Stress perturbations adjacent to salt bodies in the Deepwater Gulf of Mexico. In: SPE Annual Technical Conference and Exhibition, 5–8 October 2003, Denver, CO

  23. Hanniel, I., Elber, G.: Computing the Voronoi cells of planes, spheres and cylinders in \({\mathbb{R}}^3\). Comput. Aided Geom. Des. 26(6), 695–710 (2009)

    Article  Google Scholar 

  24. Hasan, M., Gavrilova, M.L.: A geometric approach to drill path collision avoidance. In: Proceedings of the 2010 International Symposium on Voronoi Diagrams in Science and Engineering, pp. 244–253

  25. Hoff, K.E., Keyser, J., Lin, M., Manocha, D., Culver, T.: Fast computation of generalized Voronoi diagrams using graphics hardware. In: Proceedings of the ACM SIGGRAPH Computer Graphics, Vol. 33 (Annual Conference Series), pp. 277–286 (1999)

  26. Hemmer, M., Setter, O., Halperin, D.: Constructing the exact Voronoi diagram of arbitrary lines in three-dimensional space. In: de Berg, M., Meyer, U. (Eds.) Algorithms-ESA: ESA 2010, Lecture Notes in Computer Science, Vol. 6346. Springer, Berlin (2010)

  27. Hsieh, H.-H., Tai, W.-K.: A simple GPU-based approach for 3d Voronoi diagram construction and visualization. Simul. Model. Pract. Theory 13(8), 681–692 (2005)

    Article  Google Scholar 

  28. Julia Programming Language. https://julialang.org (2019)

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

    Article  Google Scholar 

  30. Klein, R.: Concrete and abstract Voronoi diagrams. In: Goos, G., Hartmanis, J. (Eds.) Lecture Notes in Computer Science. Springer (1989)

  31. Koltun, V., Sharir, M.: Three dimensional Euclidean Voronoi diagrams of lines with a fixed number of orientations. SIAM J. Comput. 32(3), 616–642 (2003)

    Article  MathSciNet  Google Scholar 

  32. Lafon, J.C., Blanc, G.: Meshing and visualization of oilfields. In: IEEE Conference on Information Visualization (IV’97)

  33. Ledoux, H.: Computing the 3D Voronoi diagram robustly: an easy explanation. In: 4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007)

  34. Liang, C., Mourrain, B., Pavone, J.P.: Subdivision methods for 2d and 3d implicit curves. In: Jüttler, B., Piene, R. (eds.) Geometric Modeling and Algebraic Geometry, pp. 199–214. Springer, Berlin (2008)

    Chapter  Google Scholar 

  35. Okabe, A., Boots, B., Sugihara, K.: Spatial Tessellations. Concepts and Applications of Voronoi Diagrams. Wiley, New York (2000)

    Book  Google Scholar 

  36. Schaefer, S., Warren, J.: Dual marching cubes: primal contouring of dual grids. In: 12th Proceedings of the Pacific Conference on Computer Graphics and Applications, 2004 (PG 2004) (2004), pp. 70–76. ISSN: 1550-4085. https://doi.org/10.1109/PCCGA.2004.1348336

  37. Seong, J.K., Cohen, E., Elber, G.: Voronoi diagram computations for planar NURBS curves. In: Proceedings of the: ACM symposium on Solid and Physical Modelling, June 02–04, 2008, Stony Brook, NY (2008)

  38. Setter, O., Sharir, M., Halperin, D.: Constructing two-dimensional Voronoï diagrams via divide-and-conquer of envelopes in space. In: Gavrilova, M.L., Tan, C.J.K. (Eds.) Transactions on Computational Science IX, pp. 1–27. Springer (2010)

  39. Teichmann, M., Teller, S.: Polygonal approximation of Voronoi diagrams of a set of triangles in three dimensions. Technical Report 766, Laboratory of Computer Science, MIT (1997)

  40. Thorogood, J.L., Hogg, T.W., Williamson, H.S.: Application of risk analysis methods to subsurface well collisions. SPE Drill. Eng. 6(4), 299–304 (1991). https://doi.org/10.2118/23941-PA

    Article  Google Scholar 

  41. Tsao, Y.H., Dawson, C.R., Ure, D.W.: Well collision avoidance, U.S. Patent 5 901 795, May 11 1999

  42. Vleugels, J., Overmars, M.: Approximating Voronoi diagrams of convex sites in any dimension. Int. J. Comput. Geom. Appl. 8(02), 201–221 (1998)

    Article  MathSciNet  Google Scholar 

  43. Wang, Z., Inglis, T.A.: Planning directional wells through a high density cluster of existing wells. SPE Drill. Eng. 5(4), 291–293 (1990). https://doi.org/10.2118/17594-PA

    Article  Google Scholar 

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Acknowledgements

Most of the implementation part of the work was done during research visits at the research center INRIA Sophia Antipolis with partial funding from the Simons Foundation and INRIA.

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Authors and Affiliations

  1. Département de Mathématiques, Faculté des Sciences et Techniques, Université Dan Dicko Dankoulodo de Maradi, BP: 465, Maradi, Niger

    Ibrahim Adamou

  2. INRIA, Université Côte d’Azur, 2004 route des Lucioles, BP: 93, 06902, Sophia Antipolis Cedex, France

    Bernard Mourrain

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Correspondence to Ibrahim Adamou.

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This work was partially supported by a grant from Simons Foundation and INRIA.

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Adamou, I., Mourrain, B. Computing the Topology of Voronoï Diagrams of Parallel Half-Lines. Math.Comput.Sci. 15, 859–876 (2021). https://doi.org/10.1007/s11786-021-00508-1

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  • DOI: https://doi.org/10.1007/s11786-021-00508-1

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