Skip to main content
SpringerLink
Log in
Book cover

European Symposium on Algorithms

ESA 2004: Algorithms – ESA 2004 pp 448–459Cite as

Approximate Unions of Lines and Minkowski Sums

  • Conference paper

  • 1443 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3221))

Abstract

We study the complexity of and algorithms to construct approximations of the union of lines, and of the Minkowski sum of two simple polygons. We also study thick unions of lines and Minkowski sums, which are inflated with a small disc. Let b=D/ε be the ratio of the diameter of the region of interest and the distance (or error) of the approximation. We present upper and lower bounds on the combinatorial complexity of approximate and thick unions of lines and Minkowski sums, with bounds expressed in b and the input size n. We also give efficient algorithms for the computation.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Agarwal, P.K.: Range searching. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn. ch. 36, pp. 809–837. CRC Press, Boca Raton (2004)

    Google Scholar 

  2. Chazelle, B.: Triangulating a simple polygon in linear time. Discrete Comput. Geom. 6(5), 485–524 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Daniels, K., Milenkovic, V.J.: Multiple translational containment, part i: An approximate algorithm. Algorithmica 19(1-2), 148–182 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. de Berg, M., Katz, M.J., van der Stappen, A.F., Vleugels, J.: Realistic input models for geometric algorithms. Algorithmica 34, 81–97 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Efrat, A.: The complexity of the union of (α, β)-covered objects. In: Proc. 15th Annu. ACM Sympos. Computat. Geometry (1999)

    Google Scholar 

  6. Efrat, A., Katz, M.J., Nielsen, F., Sharir, M.: Dynamic data structures for fat objects and their applications. Comput. Geom. Theory Appl. 15, 215–227 (2000)

    MATH  MathSciNet  Google Scholar 

  7. Efrat, A., Rote, G., Sharir, M.: On the union of fat wedges and separating a collection of segments by a line. Comput. Geom. Theory Appl. 3, 277–288 (1993)

    MATH  MathSciNet  Google Scholar 

  8. Efrat, A., Sharir, M.: On the complexity of the union of fat objects in the plane. Discrete Comput. Geom. 23, 171–189 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Erickson, J.: Local polyhedra and geometric graphs. In: Proc. 19th Annu. ACM Sympos. Computat. Geometry, pp. 171–180 (2003)

    Google Scholar 

  10. Gajentaan, A., Overmars, M.H.: On a class of O(n2) problems in computational geometry. Comput. Geom. Theory Appl. 5, 165–185 (1995)

    MATH  MathSciNet  Google Scholar 

  11. Heckmann, R., Lengauer, T.: Computing upper and lower bounds on textile nesting problems. In: Díaz, J. (ed.) ESA 1996. LNCS, vol. 1136, pp. 392–405. Springer, Heidelberg (1996)

    Google Scholar 

  12. Kedem, K., Livne, R., Pach, J., Sharir, M.: On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Discrete Comput. Geom. 1, 59–71 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  13. Latombe, J.-C.: Robot Motion Planning. Kluwer Ac. Pub., Boston (1991)

    Google Scholar 

  14. Matoušek, J., Pach, J., Sharir, M., Sifrony, S., Welzl, E.: Fat triangles determine linearly many holes. SIAM J. Comput. 23, 154–169 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. Milenkovic, V.J.: Multiple translational containment, part ii: Exact algorithm. Algorithmica 19(1-2), 183–218 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Overmars, M.H., van der Stappen, A.F.: Range searching and point location among fat objects. Journal of Algorithms 21, 629–656 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. Pach, J., Sharir, M.: On the boundary of the union of planar convex sets. Discrete Comput. Geom. 21, 321–328 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Pach, J., Tardos, G.: On the boundary complexity of the union of fat triangles. SIAM J. Comput. 31, 1745–1760 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Sharir, M.: Algorithmic motion planning. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn. ch. 47, pp. 1037–1064. CRC Press, Boca Raton (2004)

    Google Scholar 

  20. van der Stappen, A.F., Overmars, M.H.: Motion planning amidst fat obstacles. In: Proc. 10th Annu. ACM Sympos. Comput. Geom., pp. 31–40 (1994)

    Google Scholar 

  21. van Kreveld, M., Speckmann, B.: Cutting a country for smallest square fit. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 91–102. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  22. van Kreveld, M.: On fat partitioning, fat covering, and the union size of polygons. Comput. Geom. Theory Appl. 9(4), 197–210 (1998)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Information and Computing Sciences, Utrecht University, PO Box 80.089, 3508 TB, Utrecht, The Netherlands

    Marc van Kreveld & A. Frank van der Stappen

Authors
  1. Marc van Kreveld
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Frank van der Stappen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Freiburg, Georges Köhler Allee 79, 79110, Freiburg, Germany

    Susanne Albers

  2. Department of Computer Science, King’s College, WC2R 2LS, London, UK

    Tomasz Radzik

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

van Kreveld, M., van der Stappen, A.F. (2004). Approximate Unions of Lines and Minkowski Sums. In: Albers, S., Radzik, T. (eds) Algorithms – ESA 2004. ESA 2004. Lecture Notes in Computer Science, vol 3221. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30140-0_41

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-30140-0_41

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-23025-0

  • Online ISBN: 978-3-540-30140-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation