Orthogonal Decomposition of Fractal Sets

Abstract

It is well known that interactive modeling of fractal sets is a very difficult task. The common constructive tool, Iterated Function Systems (IFS) fails to be of big help in this matter. On the contrary, the concept of Affine invariant Iterated Function Systems (AIFS) makes modeling partially possible, and in addition it offers natural algorithms that can generate both fractal (so non-smooth) and smooth, polynomial objects. While IFS is usually based on Cartesian rectangular coordinates, the AIFS is constructed using areal (normalized barycentric) coordinates. Here, we show the existence of the bijective transform between IFS and AIFS, and give explicit formulas in n-dimensional real spaces. Also, we point out that these mappings are based on orthogonal projections, and give characteristic examples.