Tutorial Chapter 11
WHAT IS IT?
IFS (iterated function system) is another way of generating fractals. It is based on taking a point or a figure and substituting it with several other identical ones. For example, there is a very simple way of generating the Sierpinski triangle. You can start with an equilateral triangle and substitute it with three smaller triangles.
By iterating this process you substitute each of those three triangles with three even smaller triangles continue a large number of times.
Mathematically, substituting a shape with another one is called a geometric transformation. The above example two kinds of transformations: translation (movement of triangles) and dilation (changing the size of the triangles). The third kind of transformation is rotation. It can be used to create fractals in which the self-similar parts are located at different angles. For example, to create a realistic fractal model of a tree you will need rotation for the branches. Other types of transformation, such as reflection and inversion can also be used to create a great variety of fractals. IFS make it relatively easy to create algorithms for drawing fractals. For most 2-D fractals, all you have to store is the list of all transformations with 6 parameters each:
1. horizontal movement
2. vertical movement
3. rotation of the figure’s vertical axis
4. rotation of the figure’s horizontal axis
5. stretching of the figure’s vertical axis
6. stretching of the figure’s horizontal axis
For 3-D fractals, you need additional 3 parameters for the z-axis. IFS can be used to create most fractals except for the formula fractals. For many IFS fractals, IFS is the only way they can be generated.
|Bills IFS4 Screensaver
A great fractals screensaver that displays iterated function systems
|Iterated Function Systems
A great website about iterated function systems. It has a very beautiful gallery, and even CGI programs that allow to create your own IFS fractals.
Michael F. Barnsley
This is another classic, written by the founder of IFS fractals. The entire book is mainly an in-depth analysis of this topic. Although relatively hard to read, the text is accompanied by a very large number of easily understandable diagrams.