Abstract
Self-similarity of Bernstein polynomials, embodied in their subdivision property is used for construction of an Iterative (hyperbolic) Function System (IFS) whose attractor is the graph of a given algebraic polynomial of arbitrary degree. It is shown that such IFS is of just-touching type, and that it is peculiar to algebraic polynomials. Such IFS is then applied to faster evaluation of Bézier curves and to introduce interactive free-form modeling component into fractal sets.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. F. Barnsley, Fractals Everywhere, Academic Press, 1988.
J. E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), 713–747.
R. T. Farouki, On the stability of transformations between power and Bernstein polynomial forms, Computer Aided Geometric Design 8 (1991), 29–36.
R. N. Goldman, Using degenerate Bézier triangles and tetrahedra to subdivide Bézier curves, Computer Aided Design 15 (1982), 307–311.
R. N. Goldman and D. C. Heath, Linear subdivision is strictly a polynomial phenomenon, Computer Aided Geometric Design 1 (1984), 269–278.
Lj. M. Kocić, Modification of Bézier curves and surfaces by degree elevation technique, Computer Aided Design 23 (1991), 692–699.
Lj. M. Kocić, Affine shape control of cubics, PU.M.A. Ser. B 3 (1992), 207–229.
B. B. Mandelbrot, The fractal geometry of nature, Freeman, San Francisco, 1982.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kocić, L.M. Fractals and Bernstein polynomials. Period Math Hung 33, 185–195 (1996). https://doi.org/10.1007/BF00150833
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00150833