Language-Restricted Iterated Function Systems, Koch Constructions, and L-systems

Introduction

Linear fractals can be generated using a variety of methods. This raises the question of finding equivalent methods for generating the same fractal. Several aspects of this question have been addressed in the literature. These include:

• A method for converting Koch constructions to equivalent iterated function systems (IFS's),
• Methods for converting selected classes of L-systems with geometric interpretation to extensions of IFS's, such as controlled iterated function systems (CIFS's) and mutually recursive function systems (MRFS's).

Previous course notes introduced the notion of language-restricted iterated function systems (LRIFS's) encompassing CIFS's and MRFS's, and included a number of sample LRIFS's equivalent to L-systems with turtle interpretation. The present notes include the following further extensions to these results:

• Introduction of the notions of iterated transformations of coordinate systems (ITCS's) and their language-restricted generalization (LRITCS's),
• A characterization of Koch constructions in terms of ITCS's,
• A method for constructing an LRITCS equivalent to a given LRIFS,
• Expression of LRITCS's using parametric L-systems with turtle interpretation.

Readers are advised that this work is still in progress, and consequently the results are not presented with the rigor expected in final publications.

Reference

Przemyslaw Prusinkiewicz and Mark Hammel. Language-restricted iterated function systems, koch constructions, and L-systems. In New Directions for Fractal Modeling in Computer Graphics, SIGGRAPH '94 Course Notes. ACM Press, 1994.