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Modeling fractal shapes using generalisations of IFS techniques

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Fractals in Engineering

Summary

One of the major problems in geometric modeling is the control of shape construction. Indeed, one should be able to construct geometrical forms by combining or manipulating simple entities. This problem is even more important when we deal with fractal geometry. In this paper, we propose some methods for increasing the modeling capabilities of fractal shape constructions. We propose two extensions of the IFS model. The first is based on the integration of free form techniques in an IFS formalism, while the second deals with the definition of matrix of attractors to give a constructive approach of fractal shapes.

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References

  1. Bandt, C. (1989): Topological Markov Chaïne and Mixed Self-Similar Sets. Math. Nachr., (142), 107–123.

    Article  MathSciNet  MATH  Google Scholar 

  2. Barnsley, M.F. (1988): Fractal Everywhere. Academic press, INC.

    Google Scholar 

  3. Berstel, J., Morcrette, M. (1989): Compact Representation of Patterns by Finite Automata. In Proceedings of Pixim, 387–395.

    Google Scholar 

  4. Culik II, K., Dube, S. (1993): Rationnal and affine expressions for image synthesis. Discrete Appl. Math., 41, 85–120, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  5. Dekking, F.M. (1980): Variations on Peano. Nieuw Archief Voor Wiskunde, (XXVIII), 275–281.

    MathSciNet  Google Scholar 

  6. Dekking, F.M. (April 1982): Recurrent sets. Advances in Mathematics, 44(1), 78–104.

    Article  MathSciNet  MATH  Google Scholar 

  7. Dubuc, S. (1987): Modèles de courbes irrégulières. In Dimensions Non Entières et Applications. Masson.

    Google Scholar 

  8. Gentil, C. (March 1992): Les fractales en synthese d’images: le modele IFS. These de doctor at, Université LYON I.

    Google Scholar 

  9. Gentil, C., Tosan, E. (December 1992): Description of fractals with IFS. In COMPUGRAPHICS’92, 214–221, Lisbon, Portugal. Portuguese ACM Chapter.

    Google Scholar 

  10. Hart, J. (May 1992): The object instancing paradigm for linear fractal modeling. In Proceedings of Graphics Interface’92.

    Google Scholar 

  11. Hata, M. (1991): Topological aspects of self-similar sets and singular functions. In J. Belair and S. Dubuc, editors, Fractal Geometry and Analysis, 255–276. Kluwer Academic Publishers.

    Google Scholar 

  12. Hwang, S.C., Yang, H.S. (1993): Discrete Approximation of the Koch Curve. Computer & Graphics, 17(1), 95–102.

    Article  Google Scholar 

  13. Lévy Véhel, J. (1988): Analyse et synthèse d’objets bi-dimensionnels par des méthodes stochastiques. Thèse de doctorat, Université de Paris-sud centre d’Orsay, INRIA.

    Google Scholar 

  14. Massopust, P. (April 1990): Fractal surfaces. Journal of Mathematical Analysis and Applications, (151), 275–290.

    Article  MathSciNet  MATH  Google Scholar 

  15. Massopust, P. (1994): Fractal function and wavelets. Academic Press.

    Google Scholar 

  16. Morcrette, M. (1996): Sur l’équivalence de descriptions de figures itérées. À pararaître dans Theoritical Computer Science.

    Google Scholar 

  17. Micchelli, CA., Prautzsch, H. (1987): Computing surfaces invariant under subdivision. Computer Aided Geometric Design, (4), 321–328.

    Article  MathSciNet  MATH  Google Scholar 

  18. Micchelli, C.A., Prautzsch, H. (1989): Union Refinement of Curves. Linear Algebra and its Applications, (114–115), 841–870.

    Article  MathSciNet  Google Scholar 

  19. Merzenich, W., Staiger, L. (1994): Fractals, dimension, and formal languages. Theoritical Informatics and Application, 28(3–4), 361–386.

    MathSciNet  MATH  Google Scholar 

  20. Mauldin, R.D., Williams, L.C. (1988): Hausdorff dimension in graph directed constructions. Trans. Amer. Math. Soc., 309, 811–829.

    Article  MathSciNet  MATH  Google Scholar 

  21. Prusinkiewicz, P., Hammel, M.S. (1991): Automata, languages and iterated function systems. In lecture notes for the SIGGRAPH’91 course: “Fractal modeling in 3D computer graphics and imagery”.

    Google Scholar 

  22. Prusinkiewicz, P., Hammel, M.S. (May 1992): Escape-time Visualization Method for Language-restricted Iterated Function Systems. In Proceedings of Graphics Interface’92.

    Google Scholar 

  23. Peitgen, H.O., Jürgens, H., Saupe, D. (1992): Encoding images by simple transformations. In Fractals For The Classroom, New York.

    Google Scholar 

  24. Prusinkiewicz, P., Sandness, G. (Novembre 1988): Koch curves as attractors and repellers. IEEE Computer Graphics & Applications, 27–40.

    Google Scholar 

  25. Thollot, J. (September 1996): Extension du modele IFS pour une geometrie fractale constructive. Thèse de doctorat, Université Claude Bernard Lyon 1.

    Google Scholar 

  26. Tosan, E. (September 1996): Une Algèbre de Formes fractales utilisant les IFS. These de doctorat, Université Claude Bernard Lyon 1.

    Google Scholar 

  27. Tricot, C. (1993): Courbes et dimension fractale. Springer Verlag.

    MATH  Google Scholar 

  28. Thollot, J., Tosan, E. (December 1993): Construction of fractales using formal languages and matrices of attractors. In H.P. Santos, editor, COMPU-GRAPHICS’93, 74–81, Technical University of Lisbon.

    Google Scholar 

  29. Thollot, J., Tosan, E. (May 1995): Constructive Fractal Geometry: constructive approach to fractal modeling using language operations. In GRAPHICS INTERFACE’95, Quebec, Canada, 196–203.

    Google Scholar 

  30. van Overveld, C.W.A.M. (1990): Family of recursively defined curves, related to the cubic Bézier curve. Computer-Aided Design, 22(9), 591–597.

    Article  MATH  Google Scholar 

  31. Zair, C.E., Tosan, E.: Computer Aided Geometric Design with IFS techniques. In FRACTALS’97, Denver, USA, April, to appear.

    Google Scholar 

  32. Zair, C.E., Tosan, E. (November 1994): Définition de courbes et surfaces lisses ou fractales à l’aide d’IFS. In AFIG 94, Toulouse, 189–200.

    Google Scholar 

  33. Zair, C.E., Tosan, E. (August 1996): Fractal modeling using free form techniques. Computer Graphics Forum, 15(3), 269–278. EUROGRAPHICS’96 Conference issue.

    Google Scholar 

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© 1997 Springer-Verlag London Limited

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Thollot, J., Zair, C.E., Tosan, E., Vandorpe, D. (1997). Modeling fractal shapes using generalisations of IFS techniques. In: Lévy Véhel, J., Lutton, E., Tricot, C. (eds) Fractals in Engineering. Springer, London. https://doi.org/10.1007/978-1-4471-0995-2_6

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  • DOI: https://doi.org/10.1007/978-1-4471-0995-2_6

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-1253-2

  • Online ISBN: 978-1-4471-0995-2

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