“Generation and display of geometric fractals in 3-D” by Norton

  • ©Alan V. Norton




    Generation and display of geometric fractals in 3-D



    We present some straightforward algorithms for the generation and display in 3-D of fractal shapes. These techniques are very general and particularly adapted to shapes which are much more costly to generate than to display, such as those fractal surfaces defined by iteration of algebraic transformations. In order to deal with the large space and time requirements of calculating these shapes, we introduce a boundary-tracking algorithm particularly adapted for array-processor implementation. The resulting surfaces are then shaded and displayed using z-buffer type algorithms. A new class of displayable geometric objects, with great diversity of form and texture, is introduced by these techniques.


    1. Artzy, E., Friedan, G., and Herman, G., The Theory, Design, Implementation and Evaluation of a Three-dimensional Surface-Detection Algorithm, Comp. Graph. & Im. Proc., 15, 1981, pp. 1-24.
    2. Carpenter, L.C., Fournier, A., and Fussel, D., Display of fractal curves and surfaces, to appear, Comm. ACM.
    3. Hamilton, Sir W. R., Elements of Quaternions, Vols. I and II, Reprinted by Chelsea Publ. Co., New York, 1969.
    4. Julia, G., Mémoire sur l’iteration des fonctions rationnelles, J. Math. Pure Appl. 4:47-245, 1918. Reprinted in Oeuvres de Gaston Julia, Vol. I, Gautier-Villars, Paris, 1968.
    5. Lin, H. K., Two and Three Dimensional Boundary Detection, Comp. Graph. and Im. Proc., 6, 1977, pp. 123-134.
    6. Mandelbrot, B. B., Fractals: Form, Chance, and Dimension, Freeman, San Francisco, 1977.
    7. Mandelbrot, B. B., Fractal Aspects of the Iteration of z→&lgr;z(1−z) for complex &lgr; and z, Ann. N. Y. Acad. Sci. 357, 1980, pp. 249-259.
    8. Mandelbrot, B. B., The Fractal Geometry of Nature, Freeman, San Francisco, 1982.
    9. Mandelbrot, B. B., and Norton, A., Fractal Surfaces Defined by Iteration of Rational Functions in the Quaternions, to appear.
    10. Newman, W. M., and Sproull, R. F., Principles of Interactive Computer Graphics, McGraw-Hill, New York, 1973.
    11. Sutherland, I., Sproull, R., and Schumacker, R., A Characterization of Ten Hidden-Surface Algorithms, Computing Surveys, Vol. 6, No. 1, 1974.

ACM Digital Library Publication: