“Construction of fractal objects with iterated function systems” by Demko, Hodges and Naylor

  • ©Stephen Demko, Laurie Hodges, and Bruce F. Naylor




    Construction of fractal objects with iterated function systems



    In computer graphics, geometric modeling of complex objects is a difficult process. An important class of complex objects arise from natural phenomena: trees, plants, clouds, mountains, etc. Researchers are at present investigating a variety of techniques for extending modeling capabilities to include these as well as other classes. One mathematical concept that appears to have significant potential for this is fractals. Much interest currently exists in the general scientific community in using fractals as a model of complex natural phenomena. However, only a few methods for generating fractal sets are known. We have been involved in the development of a new approach to computing fractals. Any set of linear maps (affine transformations) and an associated set of probabilities determines an Iterated Function System (IFS). Each IFS has a unique “attractor” which is typically a fractal set (object). Specification of only a few maps can produce very complicated objects. Design of fractal objects is made relatively simple and intuitive by the discovery of an important mathematical property relating the fractal sets to the IFS. The method also provides the possibility of solving the inverse problem. given the geometry of an object, determine an IFS that will (approximately) generate that geometry. This paper presents the application of the theory of IFS to geometric modeling.


    1. Aono, Masaki and Tosiyasu L. Kunii, “Botanical Tree Image Generation”, IEEE Computer Graphics and Applications, 4 (5), pp.10-33, (May 1984).
    2. Barnsley, Michael F. and Andrew N. Harrington, “A Mandelbrot Set for Pairs of Linear Maps” to appear in Physica 14D, (1985).
    3. Barnsley, Michael F. and Stephen Demko, “Rational Approximation and Interpolation”, Proceedings of the Tampa Conference on Rational Approximation, edited by P.R. Graves-Morris, E.B. Saff, and R.S. Varga, Springer Verlag Lecture Notes in Mathematics, No. 1105, pp. 73-88 (1984).
    4. Barnsley, Michael F. and Stephen Demko, “Iterated Function Systems and the Global Construction of Fractals”, to appear in The Proceedings of the Royal Society, preprint available, School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332.
    5. Barnsley, Michael F., “Fraetal Interpolation”, preprint available, School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332.
    6. Barnsley, Michael F., Vincent Ervin, Doug Hardin, and John Lancaster, “Solution of an Inverse Problem for Fractals and Other Sets”, pr6print available, School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, 30332.
    7. Blinn, James F., “Simulation of Wrinkled Surfaces”, Computer Graphics 12 (3), pp. 286-292 (Aug. 1978). SIG- GRAPH ’78 Proceedings.
    8. Catmull, Ed, “Computer Display of Curved Surfaces”, Proc. IEEE Conference on Computer Graphics, Pattern Recognition and Data Structure, (May 1975).
    9. Csuri, C., R. Hackathorn, R. Parent, W. Carlson, and M. Howard, “Towards an Interactive High Visual Complexity Animation System,” Computer Graphics 13 (2), pp. 289-299 (Aug. 1979). SIGGRAPH ’79 Proceedings.
    10. Demko, Stephen, Laurie Hodges, and Bruce Naylor, “Application of Iterated Function Systems to Geometric Modeling”, Technical Report GIT-ICS 85/14.
    11. Diaconis, Persi and M. Shahshahani, “Products of Random Matrices and Computer Image Generation”, Stanford University preprint.
    12. Fournier, Alain, “Stochastic Modeling in Computer Graphics”, PhD thesis, Univ. of Texas at Dallaz, Richardson, Texas, (Aug. 1980).
    13. Fournier, Alain, Don Fussell, and Loren Carpenter, “Computer Rendering of Stochastic Models”, Communications of the A CM, 25 (6) (June 1982).
    14. Fu, K.S. and S.Y. Lu, “Computer Generation of Texture Using a Syntactic Approach”, Computer Graphics 12 (3), pp. 147-152 (Aug. 1978). SIGGRAPH ’78 Proceedings.
    15. Gardner, Geoffrey Y., “Simulation of Natural Scenes Using Quadric Surfaces”, Computer Graphics 18 (3), pp.ll-20 (July 1984). SIGGRAPH ’84 Proceedings.
    16. John Hutchinson, “Fractals and Self-similarity”, Indiana University Journal of Mathematics, 30 , pp. 713-747 (1981).
    17. Mandelbrot, Benoit, The Fractal Geometry of Nature W. H. Freeman and Co., San Francisco, (1983).
    18. Reeve~, William T., “Particle Systems- A Technique for Modeling a Cla~s of Fuzzy Objects”, A CM Transactions on Graphics, 2 (2), pp.91-108 (April 1983).
    19. Smith, Airy Ray, “Plants, Fractals, and Formal Languages”, Computer Graphics 18 (a), pp. 1-10 (July 1984). SIGGRAPH ’84 Proceedings.

ACM Digital Library Publication:

Overview Page: