Fractal Shapes Description with Parametric L-systems and Turtle Algebra

Authors: Ikbal Zammouri, Béchir Ayeb

Abstract:

In this paper, we propose a new method to describe fractal shapes using parametric l-systems. First we introduce scaling factors in the production rules of the parametric l-systems grammars. Then we decorticate these grammars with scaling factors using turtle algebra to show the mathematical relation between l-systems and iterated function systems (IFS). We demonstrate that with specific values of the scaling factors, we find the exact relationship established by Prusinkiewicz and Hammel between l-systems and IFS.

Keywords: Fractal shapes, IFS, parametric l-systems, turtlealgebra.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1084770

References:

[1] Alvy Ray smith, "Plants, fractals and formal languages", Proceedings of the 11th Annual Conference on Graohics and Interactive Technics, pages 1-10, 1984, ISSN0097-8930.
[2] E. Tosan, P. Maillot, J. Azencot, B. Décoret, "On process for the construction of scenes: an attempt of formalization", Proceedings of IFIP CAPE-86, North-Holland, P382-395 (1986).
[3] E. Tosan, "Wire frame fractal topology and IFS morphisms", 4th Conference Fractals in Engeneering, Delft, The Netherlands, 14-16 June 1999, organization INRIA and TUDelft, pages 67-81.
[4] M. F. Barnsley, "Fractals everywhere", Morgan Kaufmann, 1993.
[5] P. Prusinkiewicz and A. Lindenmayer, "The algorithmic beauty of plants", Springer Verlag, 1989.
[6] P. Prusinkewicz and J. Hanan, "Visualization of botanical structures and processes using parametric l-systems",Scientific Visualization and Graphic Simulation, pages 183-201, 1990.
[7] P. Prusinkiewicz and M. Hammel, "Automata, languages and iterated function systems", Lecture Notes for the SIGGRAPH -91 course: Fractal Modeling in 3D Computer Graphics and Imagery, 1991.
[8] P. Prusinkiewicz, In M. Novak (ed.), "Self-similarity in plants: integrating mathematical and biological perspectives, thinking in patterns", Fractals and Related Phenomena in Nature, pages. 103-118, 2004.
[9] P. Prusinkiewicz, F. Samavati, C. Smith and R. Karwowski, "L-system description of subdivision curves", International Journal of Shape Modeling 9 (1), pages 41-59, 2003.
[10] R. Goldman, S. Shaefer and Tao Ju,, "Turtle geometry in computer graphics and computer aided design", Computer Aided Design 36(4): 1471-1482 (2004).
[11] T. Ju, S. Shaefer, R.Goldman, "Recursive turtle programs and iterated affine transformations", Computer & Graphics 28(6): 991-1004 (2004).
[12] T. Jo├½lle., « Extension du modèle IFS pour une géométrie fractale constructive », Phd These, Claude Bernard University of LYON1, 9 September 1996.
[13] Z. Chems Eddine, (44), «Formes fractales ├á p├┤les basées sur une généralisation des IFS », Phd These, Claude Bernard University of Lyon 1, June 1998.
[14] Z. Ikbal and A. Béchir, "A description of a family of iterated functions systems with parametric l-systems", Proceedings of IEEE Sciences of Electronics, Technologies of Information and Telecommunications, pages 1-7, Hammamet, 25-29 March 2007, Tunisia.