1. D. Geometry of Plane Transformations

To generate all but the simplest fractals, we need to understand the geometry of plane transformations. Here we describe and illustrate the four features of plane transformations, and show how they are encoded in our software.

Affine transformations of the plane are composed of scalings, reflections, rotations, and translations. Here is the matrix formulation of an affine transformation. The order of these transformations is important. For example, reflecting then translating can give a different result from translating then reflecting.

We adopt this convention:

scalings first, reflections second, rotations third, and translations last.

This order is imposed by the matrix formulation.

Emphasizing this order, the components of a transformation are encoded in tables of this form

With this encoding of transformations of the plane, we can make fractals using the method called Iterated Function Systems (IFS).