Looking around my hard disk, I found a program I wrote in December 1997 to demonstrate the capabilities of IFS (Iterated Function Series) using Affine maps. Here is an extract from the documentation I wrote at the time.
(The source code is available on request to anyone who’s interested.)
A shape, an IFS fractal (Iterated Function Series), is defined by a number of transforms. Each one of these transforms map from the whole shape to a smaller self-similar part of it. In an affine IFS fractal, each one of these transforms is affine.
An affine transform is a linear transform with translation. Examples of 2D affine transforms are rotation about an arbitrary point, reflection in an arbitrary line, translation by any amount, etc.
So an affine IFS shape can be described by a set of transforms. An implementation generally has probabilities associated with these.
Given a point on the object, the point transformed by any of the transforms will also be on the object. The algorithm works by this method. Given a point on the object, a transform is chosen according to the probabilities attached to them and the point is transformed. The new point is then marked as having been visited. The process is repeated until some bound on the number of points to be plotted has been reached. The initial point is chosen at random: after a few transforms the point will lie on the object. Hence the first couple of points found are not plotted.