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Computational and Structural Advantages of Circular Boundary Representation

  • Oswin Aichholzer
  • Franz Aurenhammer
  • Thomas Hackl
  • Bert Jüttler
  • Margot Oberneder
  • Zbyněk Šír
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4619)

Abstract

Boundary approximation of planar shapes by circular arcs has quantitive and qualitative advantages compared to using straight-line segments. We demonstrate this by way of three basic and frequent computations on shapes – convex hull, decomposition, and medial axis. In particular, we propose a novel medial axis algorithm that beats existing methods in simplicity and practicality, and at the same time guarantees convergence to the medial axis of the original shape.

Keywords

Convex Hull Voronoi Diagram Computational Geometry Medial Axis Steiner Point 
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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References

  1. 1.
    Alt, H., Cheong, O., Vigneron, A.: The Voronoi diagram of curved objects. Discrete & Computational Geometry 34, 439–453 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Attali, D., Boissonnat, J.-D., Edelsbrunner, H.: Stability and computation of medial axes – a state-of-the-art report. In: Mller, T., Hamann, B., Russell, B. (eds.) Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration, Springer Series on Mathematics and Visualization (to appear)Google Scholar
  3. 3.
    Avis, D., Toussaint, G.T.: An efficient algorithm for decomposing a polygon into star-shaped polygons. Pattern Recognition 13, 395–398 (1981)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bhattacharya, B.K., El Gindy, H.: A new linear convex hull algorithm for simple polygons. IEEE Trans. Information Theory IT-30, 85–88 (1984)CrossRefGoogle Scholar
  5. 5.
    Chazal, F., Lieutier, A.: Stability and homotopy of a subset of the medial axis. In: Proc. 9th ACM Symp. Solid Modeling and Applications, pp. 243–248 (2004)Google Scholar
  6. 6.
    Chazal, F., Soufflet, R.: Stability and finiteness properties of medial axis and skeleton. J. Dynamical and Control Systems 10, 149–170 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chazelle, B.: A theorem on polygon cutting with applications. In: Proc. 23rd IEEE Symp. FOCS, pp. 339–349 (1982)Google Scholar
  8. 8.
    Chin, F., Snoeyink, J., Wang, C.A.: Finding the medial axis of a simple polygon in linear time. In: Staples, J., Katoh, N., Eades, P., Moffat, A. (eds.) ISAAC 1995. LNCS, vol. 1004, pp. 382–391. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  9. 9.
    Choi, H.I., Choi, S.W., Moon, H.P.: Mathematical theory of medial axis transform. Pacific J. Mathematics 181, 57–88 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dobkin, D.P., Souvaine, D.L.: Computational geometry in a curved world. Algorithmica 5, 421–457 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Emiris, I.Z., Kakargias, A., Pion, S., Teillaud, M., Tsigaridas, E.P.: Towards an open curved kernel. In: Proc. 20th Ann. ACM Symp. Computational Geometry, pp. 438-446 (2004)Google Scholar
  12. 12.
    Farin, G.: Curves and Surfaces for Computer Aided Geometric Design. Academic Press, San Diego (1997)zbMATHGoogle Scholar
  13. 13.
    Farin, G., Hoschek, J., Kim, M.-S.: Handbook of Computer Aided Geometric Design. Elsevier, Amsterdam (2002)zbMATHGoogle Scholar
  14. 14.
    Garey, M.R., Johnson, D.S., Preparata, F.P., Tarjan, R.E.: Triangulating a simple polygon. Information Processing Letters 7, 175–179 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Graham, R.L.: An efficient algorithm for determining the convex hull of a finite planar set. Information Processing Letters 1, 132–133 (1972)zbMATHCrossRefGoogle Scholar
  16. 16.
    Graham, R.L., Yao, F.F.: Finding the convex hull of a simple polygon. J. Algorithms 4, 324–331 (1984)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Held, M., Eibl, J.: Biarc approximation of polygons with asymmetric tolerance bands. Computer-Aided Design 37, 357–371 (2005)CrossRefGoogle Scholar
  18. 18.
    Hertel, S., Mehlhorn, K.: Fast triangulation of the plane with respect to simple polygons. Information & Control 64, 52–76 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Klein, R., Mehlhorn, K., Meiser, S.: Randomized incremental construction of abstract Voronoi diagrams. Computational Geometry: Theory and Applications 3, 157–184 (1993)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Kong, X., Everett, H., Toussaint, G.T.: The Graham scan triangulates simple polygons. Pattern Recognition Letters 11, 713–716 (1990)zbMATHCrossRefGoogle Scholar
  21. 21.
    Lee, D.T.: Medial axis transformation of a planar shape. IEEE Trans. Pattern Analysis and Machine Intelligence PAMI-4, 363–369 (1982)CrossRefGoogle Scholar
  22. 22.
    Lee, D.T., Preparata, F.P.: Location of a point in a planar subdivision and its applications. SIAM J. Computing 6, 594–606 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    McCallum, D., Avis, D.: A linear algorithm for finding the convex hull of a simple polygon. Information Processing Letters 9, 201–206 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Meek, D.S., Walton, D.J.: Approximation of a planar cubic Bézier spiral by circular arcs. J. Computational and Applied Mathematics 75, 47–56 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Meek, D.S., Walton, D.J.: Spiral arc spline approximation to a planar spiral. J. Computational and Applied Mathematics 107, 21–30 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Melkman, A.: On-line construction of the convex hull of a simple polygon. Information Processing Letters 25, 11–12 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Ong, C.J., Wong, Y.S., Loh, H.T., Hong, X.G.: An optimization approach for biarc curve fitting of B-spline curves. Computer-Aided Design 28, 951–959 (1996)CrossRefGoogle Scholar
  28. 28.
    Ramamurthy, R., Farouki, R.T.: Voronoi diagram and medial axis algorithm for planar domains with curved boundaries I. Theoretical foundations. J. Computational and Applied Mathematics 102, 119–141 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Reif, U.: Uniform B-spline approximation in Sobolev spaces. Numerical Algorithms 15, 1–14 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Sabin, M.A.: The use of circular arcs to form curves interpolated through empirical data points. Rep. VTO/MS/164, British Aircraft Corporation (1976)Google Scholar
  31. 31.
    Šír, Z., Feichtinger, R., Jüttler, B.: Approximating curves and their offsets using biarcs and Pythagorean hodograph quintics. Computer-Aided Design 38, 608–618 (2006)CrossRefGoogle Scholar
  32. 32.
    Yang, X.: Efficient circular arc interpolation based on active tolerance control. Computer-Aided Design 34, 1037–1046 (2002)CrossRefGoogle Scholar
  33. 33.
    Yap, C.K.: An O(n logn) algorithm for the Voronoi diagram of a set of simple curve segments. Discrete & Computational Geometry 2, 365–393 (1987)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Oswin Aichholzer
    • 1
  • Franz Aurenhammer
    • 1
  • Thomas Hackl
    • 1
  • Bert Jüttler
    • 2
  • Margot Oberneder
    • 2
  • Zbyněk Šír
    • 2
  1. 1.University of Technology GrazAustria
  2. 2.Johannes Kepler University of LinzAustria

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