Computational Geosciences

, Volume 16, Issue 4, pp 853–869 | Cite as

Walking algorithms for point location in TIN models

  • Roman Soukal
  • Martina Málková
  • Ivana Kolingerová
Review Paper
First Online:
Received:
Accepted:

Abstract

Finding which triangle in a planar or 2.5D triangle mesh contains a query point (so-called point location problem) is a frequent task in geosciences, especially when working with triangulated irregular network models. Usually, a large number of point locations has to be performed, and so there is a need for fast algorithms having minimal additional memory requirements and resistant to changes in the triangulation. So-called walking algorithms offer low complexity, easy implementation, and negligible additional memory requirements, which makes them suitable for such applications. In this article, we focus on these algorithms, summarize, and compare them with regard to their use in geosciences. Since such a summary has not been done yet, our article should serve those who are dealing with this problem in their application to decide which algorithm would be the best for their solution. Moreover, the influence of the triangulation type on the number of the visited triangles is discussed.

Keywords

Point searching Searching algorithms Planar triangulation TIN models Terrain models 
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References

  1. 1.
    Amenta, N., Choi, S., Rote, G.: Incremental constructions con BRIO. In: SCG ’03: Proceedings of the 19th Annual Symposium on Computational Geometry, pp. 211–219. ACM, New York, NY, USA (2003)Google Scholar
  2. 2.
    Beirouti, R.: A fast heuristic for finding the minimum weight triangulation. Tech. Rep., Vancouver, BC, Canada (1997)Google Scholar
  3. 3.
    Boissonnat, J.D., Teillaud, M.: On the randomized construction of the Delaunay tree. Theor. Comp. Sci. 112(2), 339–354 (1993)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Buchin, K.: Incremental construction along space-filling curves. In: EuroCG’05: Proceedings of the 21th European Workshop on Computational Geometry, pp. 17–20 (2005)Google Scholar
  5. 5.
    Dæhlen, M., Fimland, M., Hjelle, Ø.: A Triangle-based Carrier for Geographical Data, pp. 105–120. Taylor and Francis, New York (2001)Google Scholar
  6. 6.
    Devillers, O.: The Delaunay hierarchy. Int. J. Found. Comput. Sci. 13, 163–180 (2002)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Devillers, O., Pion, S., Teillaud, M.: Walking in a triangulation. In: Proceedings of the 17th Annual Symposium on Computational Geometry, pp. 106–114 (2001)Google Scholar
  8. 8.
    Devroye, L., Mucke, E.P., Zhu, B.: A note on point location in Delaunay triangulations of random points. Algorithmica 22(4), 477–482 (1998)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Drysdale, R.L.S., Rote, G., Aichholzer, O.: A simple linear time greedy triangulation algorithm for uniformly distributed points. Report IIG-408, Institutes for Information Processing. Technische Universit at Graz (1995)Google Scholar
  10. 10.
    Floriani, L.D., Falcidieno, B., Nagy, G., Pienovi, C.: On sorting triangles in a Delaunay tessellation. Algorithmica 6(4), 522–532 (1991)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Green, P.J., Sibson, R.: Computing Dirichlet tessellations in the plane. Comput. J. 21(2), 168–173 (1978)MathSciNetMATHGoogle Scholar
  12. 12.
    Hansford, D.: The neutral case for the min-max triangulation. Comput. Aided Geom. Des. 7(5), 431–438 (1990)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Höhle, J., Oster Pedersen, C., Bayer, T., Frederiksen, P.: The photogrammetric derivation of digital terrain models in built-up areas. Photogramm. J. Finl. 22(1), 33–45 (2010)Google Scholar
  14. 14.
    Koch, A.: An integrated semantically correct 2.5 dimensional object oriented TIN. In: Proceedings of the 1st International Workshop on Next Generation 3D City Models (2005)Google Scholar
  15. 15.
    Kolingerová, I.: A small improvement in the walking algorithm for point location in a triangulation. In: Proceedings of the 22nd European Workshop on Computational Geometry, pp. 221–224 (2006)Google Scholar
  16. 16.
    Lawson, C.L.: Mathematical Software III; Software for C1 Surface Interpolation, pp. 161–194. Academic, New York (1977)Google Scholar
  17. 17.
    Mehlhorn, K., Näher, S.: Leda: a platform for combinatorial and geometric computing. Commun. ACM 38(1), 96–102 (1995)CrossRefGoogle Scholar
  18. 18.
    Mücke, E.P., Saias, I., Zhu, B.: Fast randomized point location without preprocessing in two and three-dimensional Delaunay triangulations. In: Proceedings of the 12th Annual Symposium on Computational Geometry, vol. 26, pp. 274–283 (1996)Google Scholar
  19. 19.
    Mulmuley, K.: Randomized multidimensional search trees: dynamic sampling. In: Proceedings of the 7th Annual Symposium on Computational Geometry, pp. 121–131 (1991)Google Scholar
  20. 20.
    Owens, K., Parikh, R.: Fast random number generator on the Intel Pentium 4 processor. Intel Software Network (2009)Google Scholar
  21. 21.
    Pearson, K.: Notes on the history of correlation. In: Royal Society Proceedings, pp. 25–45 (1920)Google Scholar
  22. 22.
    Purchart, V., Kolingerová, I., Beneš, B.: Interactive sand-covered terrain surface model with haptic feedback. In: GIS Ostrava 2012 - Surface Models for Geosciences, pp. 215–223 (2012)Google Scholar
  23. 23.
    Schilling, A., Basanow, J., Zipf, A.: Vector based mapping of polygons on irregular terrain meshes for web 3D map services. In: 3rd International Conference on Web Information Systems and Technologies (WEBIST), Barcelona (2007)Google Scholar
  24. 24.
    Schilling, A., Lanig, S., Neis, P., Zipf, A.: Integrating terrain surface and street network for 3d routing. In: 3D Geo-Information Sciences, Lecture Notes in Geoinformation and Cartography, pp. 109–126. Springer, Berlin Heidelberg (2009)Google Scholar
  25. 25.
    Shewchuk Jonathan, R.: Adaptive precision floating-point arithmetic and fast robust geometric predicates. Discrete Comput. Geom. 18, 305–363 (1996)CrossRefGoogle Scholar
  26. 26.
    Sloan, S.W.: A fast algorithm for constructing Delaunay triangulations in the plane. Adv. Eng. Softw. 9(1), 34–55 (1987)MATHCrossRefGoogle Scholar
  27. 27.
    Soukal, R., Kolingerová, I.: Straight walk algorithm modification for point location in a triangulation. In: EuroCG’09: Proceedings of the 25th European Workshop on Computational Geometry, pp. 219–222. Brussels, Belgium (2009)Google Scholar
  28. 28.
    Soukal, R., Kolingerová, I.: Star-shaped polyhedron point location with orthogonal walk algorithm. Procedia CS 1(1), 219–228 (2010)CrossRefGoogle Scholar
  29. 29.
    Su, P., Drysdale, R.L.S.: A comparison of sequential Delaunay triangulation algorithms. In: Proceedings of the 11th Annual Symposium on Computational Geometry, pp. 61–70 (1995)Google Scholar
  30. 30.
    Sundareswara, R., Schrater, P.: Extensible point location algorithm. In: International Conference on Geometric Modeling and Graphics, pp. 84–89 (2003)Google Scholar
  31. 31.
    Vasa, L., Skala, V.: A perception correlated comparison method for dynamic meshes. IEEE Trans. Vis. Comput. Graph. 17(2), 220–230 (2011)CrossRefGoogle Scholar
  32. 32.
    Vigo, M., Pla, N., Cotrina, J.: Regular triangulations of dynamic sets of points. Comput. Aided Geom. Des. 19(2), 127–149 (2002)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Weller, F.: On the total correctness of Lawson’s oriented walk. In: Proceedings of the 10th International Canadian Conference on Computational Geometry, pp. 10–12 (1998)Google Scholar
  34. 34.
    Zadravec, M., Žalik, B.: An almost distribution independent incremental Delaunay triangulation algorithm. Vis. Comput. 21(6), 384–396 (2005)CrossRefGoogle Scholar
  35. 35.
    Zemek, M., Kolingerová, I.: Hybrid algorithm for deletion of a point in regular and delaunay triangulation. In: Proceedings of the Spring Conference on Computer Graphics, pp. 137–144. Budmerice, Slovakia (2009)CrossRefGoogle Scholar
  36. 36.
    Zhou, S., Jones, C.B.: HCPO: an efficient insertion order for incremental Delaunay triangulation. Inf. Process. Lett. 93(1), 37–42 (2005)MATHCrossRefGoogle Scholar
  37. 37.
    Zhu, B.: On lawsons oriented walk in random delaunay triangulations. In: Lingas, A., Nilsson, B. (eds.) Fundamentals of Computation Theory. Lecture Notes in Computer Science, vol. 2751, pp. 222–233. Springer, Berlin (2003)CrossRefGoogle Scholar
  38. 38.
    Žalik, B., Kolingerová, I.: An incremental construction algorithm for Delaunay triangulation using the nearest-point paradigm. Int. J. Geogr. Inf. Sci. 17(2), 119–138 (2003)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Roman Soukal
    • 1
  • Martina Málková
    • 1
  • Ivana Kolingerová
    • 1
  1. 1.Faculty of Applied Sciences, Department of Computer Science and EngineeringUniversity of West BohemiaPilsenCzech Republic

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