Advertisement

Mathematics in Computer Science

, Volume 6, Issue 3, pp 261–266 | Cite as

On Computing the Convex Hull of (Piecewise) Curved Objects

  • Franz AurenhammerEmail author
  • Bert Jüttler
Article

Abstract

We utilize support functions to transform the problem of constructing the convex hull of a finite set of curved objects into the problem of computing the upper envelope of piecewise linear functions. This approach is particularly suited if the objects are (possibly intersecting) circular arcs in the plane.

Keywords

Convex hull Support function Circular arcs 

Mathematics Subject Classification

Primary 68U05 Secondary 65D18 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aichholzer O., Alt H., Rote G.: Matching shapes with a reference point. Int. J. Comput. Geom. Appl. 7, 349–363 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Aichholzer O., Aurenhammer F., Hackl T., Jüttler B., Oberneder M., Šír Z.: Computational and structural advantages of circular boundary representation. Int. J. Computat.Geom. Appl. 21, 47–69 (2011)zbMATHCrossRefGoogle Scholar
  3. 3.
    Alt H., Cheong O., Vigneron A.: The Voronoi diagram of curved objects. Discret. Comput. Geom. 34, 439–453 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Aurenhammer F.: Power diagrams: properties, algorithms, and applications. SIAM J. Comput. 16, 78–96 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bajaj C.L., Kim M.-S.: Convex hulls of objects bounded by algebraic curves. Algorithmica 6, 533–553 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Boissonnat J.-D., Cerezo A., Devillers O., Duquesne J., Yvinec M.: An algorithm for constructing the convex hull of a set of spheres in dimension d. Comput. Geom. Theory Appl. 6, 123–130 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Boissonnat, J.-D., Delage, C.: Convex hull and Voronoi diagram of additively weighted points. In: Proceedings 13th European Symposium on Algorithms, vol. 3669, pp. 367–378, Springer LNCS (2005)Google Scholar
  8. 8.
    Boissonnat, J.-D., Karavelas, M.: On the combinatorial complexity of Euclidean Voronoi cells and convex hulls of d-dimensional spheres. In: Proceedings 14th ACM-SIAM Symposium on Discrete Algorithms, pp. 305–312, (2003)Google Scholar
  9. 9.
    Dobkin D.P., Souvaine D.L.: Computational geometry in a curved world. Algorithmica 5, 421–457 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Ghosh P.K., Kumar K.V.: Support function representation of convex bodies, its application in geometric computing, and some related representations. Comput. Vision Image Underst. 72, 397–403 (1998)CrossRefGoogle Scholar
  11. 11.
    Graham R.: An efficient algorithm for determining the convex hull of a finite point set. Inf. Process. Lett. 1, 132–133 (1972)zbMATHCrossRefGoogle Scholar
  12. 12.
    Gruber P.M., Wills J.M.: Handbook of Convex Geometry. Elsevier, North-Holland, Amsterdam (1993)Google Scholar
  13. 13.
    Jarvis R.A.: On the identification of the convex hull of a finite set of points in the plane. Inf. Process. Lett. 2, 18–21 (1973)zbMATHCrossRefGoogle Scholar
  14. 14.
    Kirkpatrick D.G., Seidel R.: The ultimate planar convex hull algorithm?. SIAM J. Comput. 15, 287–299 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Li Z., Meek D.S.: Smoothing an arc spline. Comput. Graph. 29, 576–587 (2005)CrossRefGoogle Scholar
  16. 16.
    Melkman A.: On-line construction of the convex hull of a simple polygon. Inf. Process. Lett. 25, 11–12 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Nielsen F., Yvinec M.: An output-sensitive convex hull algorithm for planar objects. Int. J. Comput. Geom. Appl. 8, 39–65 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Piegl L.A., Tiller W.: Biarc approximation of NURBS curves. Comput. Aided Des. 34, 807–814 (2002)CrossRefGoogle Scholar
  19. 19.
    Prince J.R., Willsky A.S.: Reconstructing convex sets from support line measurements. IEEE Trans. Pattern Anal. Mach. Intell. 12, 377–389 (1990)CrossRefGoogle Scholar
  20. 20.
    Rappaport D.: A convex hull algorithm for discs, and applications. Comput. Geom. Theory Appl. 1, 171–187 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Richardson T.J.: Approximation of planar convex sets from hyperplanes probes. Discret. Comput. Geom. 18, 151–177 (1997)zbMATHCrossRefGoogle Scholar
  22. 22.
    Schäffer A.A., Van Wyk C.J.: Convex hulls of piecewise-smooth Jordan curves. J. Algorithms 8, 66–94 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Sharir M., Agarwal P.K.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  24. 24.
    Yue Y., Murray J.L., Corney J.R., Clark D.E.R.: Convex hull of a planar set of straight and circular line segments. Eng. Comput. 16, 858–875 (1999)zbMATHCrossRefGoogle Scholar
  25. 25.
    Yap C.K.: An O(n log n) algorithm for the Voronoi diagram of a set of simple curve segments. Discret. Comput. Geom. 2, 365–393 (1987)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Institute for Theoretical Computer ScienceUniversity of TechnologyGrazAustria
  2. 2.Institute of Applied GeometryJohannes Kepler UniversityLinzAustria

Personalised recommendations