Efficient Ray Tracing Using Interval Analysis

  • Jorge Flórez
  • Mateu Sbert
  • Miguel A. Sainz
  • Josep Vehí
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4967)


Ray tracing of implicit surfaces suffers of reliability problems: when an Implicit Surface is ray traced without guaranteed methods, thin details of the surface are not rendered correctly. The current guaranteed methods, although simple to implement, are slower than not reliable methods. This paper shows a guaranteed method that uses interval analysis to obtain extra-information of the surfaces. This information allows to determine complicated regions in which more work has to be done to obtain antialiased images, avoiding unnecessary work in other less problematic areas. This method reduces the time required to render implicit surfaces, obtaining images with the same quality of other methods in which a regular number of rays are traced per pixel.


Intersection Test Interval Analysis Inclusion Function Inside Region 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.


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  1. 1.
    Kalra, D., Barr, A.: Guaranteed ray intersection with implicit surfaces. Computer Graphics (Siggraph proceedings) 23, 206–297 (1989)Google Scholar
  2. 2.
    Hart, J.C.: Sphere tracing: A geometric method for the antialiased ray tracing of implicit surfaces. Siggraph 93 Course Notes 12(10), 527–545 (1997)Google Scholar
  3. 3.
    Capriani, O., Hvidegaard, L., Mortensen, M., Schneider, T.: Robust and efficient ray intersection of implicit surfaces. Reliable Computing 1(6), 9–21 (2000)CrossRefGoogle Scholar
  4. 4.
    Flórez, J., Sbert, M., Sainz, M.A., Vehí, J.: Improving the interval ray tracing of implicit surfaces. In: Nishita, T., Peng, Q., Seidel, H.-P. (eds.) CGI 2006. LNCS, vol. 4035, pp. 655–664. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    de Cusatis, A., de Figueredo, L., Gatas, M.: Interval methods for ray casting implicit surfaces with affine arithmetic. In: Proceedings of SIBGRAPH (1999)Google Scholar
  6. 6.
    Mitchell, D.: Robust ray intersection with interval arithmetic. In: Proceedings on Graphics interface 1990, pp. 68–74 (1990)Google Scholar
  7. 7.
    Sanjuan-Estrada, J., Casado, L., 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.
    Whitted, T.: An improved illumination model for shaded display. Communications of the ACM 23(6), 343–349 (1980)CrossRefGoogle Scholar
  9. 9.
    Amanatides, J.: Ray tracing with cones. Computer Graphics 18(3), 129–135 (1984)CrossRefGoogle Scholar
  10. 10.
    Heckbert, P., Hanrahan, P.: Beam tracing polygonal objects. Computer Graphics 18(3), 119–127 (1984)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Jorge Flórez
    • 1
  • Mateu Sbert
    • 1
  • Miguel A. Sainz
    • 1
  • Josep Vehí
    • 1
  1. 1.Institut d’Electrònica, Informàtica i AutomàticaUniversitat de GironaGironaSpain

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