Abstract
Fractal sets manipulation and modeling is a difficult task due to their complexity and unpredictability. One of the basic problems is to determine bounds of a fractal set given by some recursive definitions, for example by an Iterated Function System (IFS). Here we propose a method of bounding an IFS-generated fractal set by a minimal simplex that is affinely identical to the standard simplex. First, it will be proved that for a given IFS attractor, such simplex exists and it is unique. Such simplex is then used for definition of an Affine invariant Iterated Function System (AIFS) that then can be used for affine transformation of a given fractal set and for its modeling.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Barnsley, M.F.: Fractals everywhere. Academic Press, London (1988)
Berger, M.A.: Random affine iterated function systems: curve generation and wavelets. SIAM Review 34, 361–385 (1992)
Dubuc, S., Elqortobi, A.: Approximation of fractal sets. J. Comput. Appl. Math. 29, 79–89 (1990)
Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5) (1981)
Kocić, L.M.: Descrete methods for visualising fractal sets. FILOMAT (Niš) 9(3), 753–764 (1995)
Kocić, L.M., Simoncelli, A.C.: Towards free-form fractal modelling. In: Daehlen, M., Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces II, pp. 287–294. Vanderbilt University Press, Nashville (1998)
Kocić, L.M., Simoncelli, A.C.: Stochastic approach to affine invariant IFS. Prague Stochastic 1998 theory, Statistical Decision Functions and Random Processes 2, 317–320 (1998)
Kocić, L.M., Simoncelli, A.C.: Cantor Dust by AIFS. FILOMAT (Niš) 15, 265–276 (2001); Math. Subj. Class. 28A80 (65D17) (2000)
Kocić, L.M., Simoncelli, A.C.: Shape predictable IFS representations. In: Novak, M.M. (ed.) Emergent Nature, pp. 435–436. World Scientific, Singapore (2002); Math. Subj. Class. 65D17, 28A80 (1991)
Kocić, L.M., Stefanovska, L., Babače, E.: Affine invariant iterated function system and minimal simplex problem. In: Proceedings of the International Conference DGDS-2007, pp. 119–128. Geometry Balkan Press, Bucharest (2008)
Lawlor, O.S., Hart, J.C.: Bounding Recursive Procedural Models using Convex Optimization. In: Proc. Pacific Graphics 2003 (October 2003)
Mandelbrot, B.: The Fractal Geometry of Nature. Freeman and Company, New York (1977)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Babače, E., Kocić, L. (2009). Minimal Simplex for IFS Fractal Sets. In: Margenov, S., Vulkov, L.G., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2008. Lecture Notes in Computer Science, vol 5434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00464-3_16
Download citation
DOI: https://doi.org/10.1007/978-3-642-00464-3_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00463-6
Online ISBN: 978-3-642-00464-3
eBook Packages: Computer ScienceComputer Science (R0)