8  IFS - Iterated Function Systems

The IFS Widget

The IFS Widget

Iterative function systems are generators for self similar images.
Further information on IFS can be found in Barnsley. This tool allows to define up to 4 affine maps per image.
Predefined sets of maps are available for

8.1 Calculate

Here are two modes available to compute the iterated system:

[Copy machine] switched off

Starting from an initial point the chaos game is started.
The number of points to be drawn can be selected.
Due to the accuracy in the random function the algorithm runs into a cycle.
Such only a few number of points can be distinguished.

[Copy machine] switched on

A quadrangle is mapped by the given affine maps.
Pressing calculate gain maps the newly created quadrangles again and so on.
Due to the increasing number of quadrangles this process becomes slower and slower.
Such only 8 iterations are possible by the machine.

Reset resets this process.

If [Color] is selected each step is plotted in a different color.

If [Clear between] is selected only the quadrangles of the last step are printed.

Reset

Resets the iteration of the copy machine.

Clear screen

All printed stuff is deleted. The maps still exist.

8.2 PS print/Print

A Postscript output is generated and stored in <Path to Store Directory>/plots.
The PS-files are named automatically.

8.3 Set IFS

A reference rectangle occurs in the picture window.
By left mouse click up to four affine maps can be defined.
The first click defines the image point of the upper left corner, the second of the lower left corner and the third the image point of the lower right corner.
The resulting quadrangle is plotted.
This process can be repeated three times.
The last map is additionally stored as fourth map. It is only evaluated if

  • four different maps are defined
    or
  • [Condensation] is switched on

8.4 Set Parameters

The IFS Widget

The IFS Widget

The parameters of the actual IFS are shown and can be modified by

  • modifying them and then pressing [Apply]
    or
  • selecting another predefined IFS
    or
  • loading an IFS which has been saved earlier
    or
  • defining a new IFS by Set IFS

In the parameter window each row represents one map.
The parameters are defined as follows: \[ f_i \left ( \begin{array}{c} x\\ y\end{array} \right ) = \left ( \begin{array}{cc} a_i&b_i\\ c_i&d_i\end{array} \right ) \\ \left ( \begin{array}{c} x\\y\end{array} \right ) + \left ( \begin{array}{c} e_i\\f_i\end{array} \right ) \; , \qquad i=1,..,4 \] The values \(w_i\) in the last column give the probabilities for selecting this map by the random game.
If the sum of the first probabilities exceeds 1, the following maps are not evaluated.
If [Condensation] is active, the last map is evaluated as condensation independent of the probabilities.
The probability of the predefined maps are defined by the determinant of the matrix.
If an IFS is defined by Set IFS the probabilities are equally distributed.
### Load & Save The values can be saved in a file. The files holding IFS-data get the extension .ifs and are stored in <Path to Mathtools>/mftfiles by default.

References

BARNSLEY

BARNSLEY, Michael F.: Fractals everywhere. Morgan Kaufmann, 2000. – 534 S

  1. The Chessboard is defined by 5 affine maps. The last one is treated as condensation!↩︎