A mapping S is called "contraction" if there is a number c with 0 < c < 1 and |S(x) - S(y)| < c.|x - y|. Now suppose S1,...,Sn are contractions, then we call a set F "invariant" if equality holds for
F = S1(F) U S2(F) U ... U Sn(F) (U = Union).
The following statements can be proved:
- For a finite number of contractions S1,...,Sn there always exists an invariant set F, which is not empty (and compact).
- F ist often what we call "fractal", i.e. it has very complex structure, it is "self-similar", it has a rather simple definition and can be generated by iteration etc.
- A pre-fractal of F (a visual approximation of F, because most of the time you can't display exactly F) can be generated using the following iteration algorithm:
1. Pick at random a starting point (Xn,Yn)
2. In each iteration, pick at random one of the contractions;
Replace the iteration-point, i.e. (Xn,Yn) = S(Xn,Yn);
Display (Xn,Yn) (after about 100 iterations);
If the user didn't abort, go on!
The interesting fact about IFS is, you can describe very complex graphics by just a few parameters. It can be shown that every compact set can be approximated by invariant sets of a finite number of contractions. This means you can take pictures and describe them through mathematical formulas, and this takes much less storage-space than conventional graphics-formats.
The program IFS Paint 1.1
There are other programs that can also display or generate IFS fractals (like FractInt & WinFract), but only IFS Paint allows you to paint them graphically, just as if you were working with CorelDraw or something similar. The advantage: you will finally "understand" these pictures everybody generates, but knows nothing about.
IFS Paint is an MDI Windows application with a lot of comfort. There are many program options available, statistics, optimizations etc. And IFS Paint is FREE!!! Yes, it is public domain! Try the program, you'll love it!
A short description of the functionality:
Create, modify & test IFS fractals; save them, also in BMP and/or RLE file format. The program uses a file format that is compatible to FractInt & WinFract (though only one-way: WinFract files allow several entries in one file and comments too, IFS Paint doesn't). Test what happens if probabilities aren't optimized for the generation of the invariant set (amazing effects!!!).
Contractions are displayed as rectangles (with an "L" in the upper left corner) that can be translated, rotated, stretched or inverted using the mouse (click on the corners of the wire frames).
Add, delete or copy contractions; load from and save to a file your favorite IFS fractals. Generate the invariant sets (the fractal graphic!) and save it.
Look at the comments in the INI file (some options are accessible only there).
If you are curious about IFS, read one ore more of the following books:
M.F.Barnsley, Fractals Everywhere, Academic Press, London 1988
M.F.Barnsley, L.P.Hurd, Fractal Image Compression, AK Peters, Wellesley 1993
ifspaint.ini INI-File, program works anyway if missing
ifspaint.txt This file
*.ifs Sample files
The author is reachable on CompuServe (100275,3153) and via InterNet (100275.3153@compuserve.com). Please report the errors you find!
If you speak german, there is a complete documentation (a WinWord-2 file with more and better explanations) with the german version of IFS Paint in the German Forum of CompuServe (GO GERNET). Source Code available at request.
IFS Paint 1.1 (C) 1993,1994 by Morad Khosrow Shahi. The author takes no responsability for the correctness of the results given by the program, nor for the correctness of the information in this file. No changes whatsoever are allowed to the program without written consent of the author. Feel free to copy and distribute the program.
BWCC.DLL is a Borland product that may be distributed with executable files created by Borland compilers.
The Graphics algorithms ("Save Bitmap As" in the file menu) are extracted from the program "Bitmaps" by Larry Widing (downloaded from CompuServe). Thank you Larry, that saved me!
