Improving the Interval Ray Tracing of Implicit Surfaces

  • Jorge Flórez
  • Mateu Sbert
  • Miguel A. Sainz
  • Josep Vehí
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4035)


This paper presents a fast and reliable method to trim non-solution regions in an interval ray tracing process. The “trimming algorithm” uses interval analysis to perform rejection tests in a set of pixels simultaneously, instead of individual pixels at each time. With this approach, the presented algorithm runs faster than the traditional interval ray tracing algorithm. Also, an interval algorithm to remove aliasing in the rendering of implicit surfaces is introduced. This algorithm obtains better visualizations than the traditional point sampling. This algorithm can render thin features that would be impossible to obtain with point sampling algorithms.


Computer Graphic Intersection Test Interval Analysis Inclusion Function Interval Arithmetic 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Moore, R.: Interval analysis. Prentice-Hall, Englewood (1966)MATHGoogle Scholar
  2. 2.
    Agrawal, A., Requicha, A.A.G.: A paradigm for the robust design of algorithms for geometric models. In: Eurographics 1994, Computer Graphics Forum, vol. 13(3), pp. 33–44 (1994)Google Scholar
  3. 3.
    Ratschek, H., Rokne, J.: Geometric computations with interval and new robust methods. Horwood Publishing (2003)Google Scholar
  4. 4.
    Kalra, D., Barr, A.: Guaranteed ray intersection with implicit surfaces. In: Computer Graphics (Siggraph proceedings), vol. 23, pp. 297–306 (1989)Google Scholar
  5. 5.
    Capriani, O., Hvidegaard, L., Mortensen, M., Schneider, T.: Robust and efficient ray intersection of implicit surfaces. Reliable Computing 6, 9–21 (2000)CrossRefGoogle Scholar
  6. 6.
    Mitchell, D.: Robust ray intersection with interval analysis. In: Proceedings on graphics interface, pp. 68–74 (1990)Google Scholar
  7. 7.
    Sanjuan-Estrada, J.F., Casado, L.G., García, I.: Reliable algorithms for ray intersection in computer graphics based on interval arithmetic. In: XVI Brazilian symposium on computer graphics and Image processing, pp. 35–44 (2003)Google Scholar
  8. 8.
    de Cusatis, L., de Figueiredo, Gatas, M.: Interval methods for ray casting implicit surfaces with affine arithmetic. In: Proceedings of SIBGRAPI 1999, pp. 65–71. IEEE Press, Los Alamitos (October 1999)Google Scholar
  9. 9.
    Heckbert, Hanrahan.: Beam Tracing Polygonal Objects. In: Proceedings of the 11th annual conference on computer graphics and interactive techniques, pp. 119–127 (1984)Google Scholar
  10. 10.
    Fundamentals of computer graphics. Peter Shirley. A K Peters Ltd. (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jorge Flórez
    • 1
  • Mateu Sbert
    • 1
  • Miguel A. Sainz
    • 1
  • Josep Vehí
    • 1
  1. 1.Institut d’Informática i AplicacionsUniversitat de GironaGironaSpain

Personalised recommendations