Advertisement

Boundary Tracking of Complicated Surfaces with Applications to 3-D Julia Sets

  • C. Zahlten
Part of the Beiträge zur Graphischen Datenverarbeitung book series (GRAPHISCHEN)

Abstract

Cross sections of Julia sets in the quaternions are highly complicated 3-dimen-sional objects, which may serve as qualified test objects for surface construction and rendering algorithms. Boundary tracking methods generate the surfaces as lists of primitives such that rotation or repositioning requires only re-rendering. The main disadvantage of early boundary tracking approaches is the amount of storage they require. Ray-tracing methods, although adapted to the fractal setting, are time consuming since the objects have to be generated anew for each rendering. The Chain of Cubes algorithm used in this article is a boundary tracking method which uses a minimum of storage to generate a polygonal approximation of an iso-valued surface.

Keywords

Periodic Point Escape Time Complicated Surface Polygonal Approximation Boundary Tracking 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [ALL85]
    EX. Allgower, P.H. Schmidt: An Algorithm for Piecewise Linear Approximation of an Implicit Defined Manifold, SIAM J. Numer. Anal. 22, 322–346, 1985MathSciNetzbMATHCrossRefGoogle Scholar
  2. [ALL87]
    EX. Allgower, S. Gnutzmann: An Algorithm for Piecewise Linear Approximation of Implicitly Defined Two-Dimensional Surfaces, SIAM J. Numer. Anal. 24, 452–469, 1987MathSciNetzbMATHCrossRefGoogle Scholar
  3. [ALL90]
    EX. Allgower, K. Georg: Introduction to Numerical Continuation Methods, Springer-Verlag, New York, 1990CrossRefGoogle Scholar
  4. [HAR89]
    J.C. Hart, D.J. Sandin, L.H. Kauffman: Ray Tracing Deterministic 3-D Fractals, Computer Graphics (Proc. SIGGRAPH) 23(3), 289–296, 1989CrossRefGoogle Scholar
  5. [JÜR89]
    H. Jürgens: Optimierte Oberflächenabtastung mit orientierten Kubusketten, in H. Jürgens, D. Saupe (Eds.): Visualisierung in Mathematik und Naturwissenschaften, 53–66, Springer-Verlag, Heidelberg, 1989CrossRefGoogle Scholar
  6. [LIC91]
    R. Lichtenberger: Visualisierung von Fraktalen mit Raytracing, Diplomarbeit, Universität Bremen, 1991Google Scholar
  7. [LOR87]
    W.E. Lorensen, H.E. Cline: Marching Cubes: a High Resolution 3D Surface Construction Algorithm, Computer Graphics (Proc. SIGGRAPH) 21(4), 163–169, 1987CrossRefGoogle Scholar
  8. [MAN82]
    B.B. Mandelbrot: The Fractal Geometry of Nature, Freeman, San Francisco, 1982zbMATHGoogle Scholar
  9. [NOR82]
    A. Norton: Generation and Display of Geometric Fractals in 3-D, Computer Graphics (Proc. SIGGRAPH) 16(3), 61–66, 1982CrossRefGoogle Scholar
  10. [NOR89]
    A. Norton: Julia Sets in the Quaternions, Computers & Graphics 13(2), 267–278, 1989CrossRefGoogle Scholar
  11. [PEI86]
    H.-O. Peitgen, P. Richter: The Beauty of Fractals, Springer-Verlag, New York, 1986zbMATHCrossRefGoogle Scholar
  12. [RUM91]
    M. Rümpf et al.: GRAPE GRAphics Programming Environment, Institut für Angewandte Mathematik der Universität Bonn, SFB 256, 1991Google Scholar
  13. [WYV86]
    G. Wyvill, C. McPheeters, B. Wyvill: Data Structure for Soft Objects, Visual Computer 2(4), 227–234, 1986CrossRefGoogle Scholar
  14. [ZAH91]
    C. Zahlten: Piecewise Linear Approximation of Isovalued Surfaces, to appear in the Proceedings of the 2nd Eurographics Workshop on Visualization in Scientific Computing, Delft, The Netherlands, 22-24 April, 1991Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • C. Zahlten
    • 1
  1. 1.Institut für Dynamische SystemeUniversität BremenGermany

Personalised recommendations