On the Geometry of IFS Fractals and its Applications

Abstract

Visually complex objects with infinitesimally fine features, naturally call for mathematical representations. The geometrical property of self-similarity - the whole similar to its parts - when iterated to infinity generates such features.

Finite sets of affine contractions called Iterated Function Systems (IFS), with their compact attractors IFS fractals, can be applied to represent detailed self-similar shapes, such as trees or mountains. The fine local features of such attractors prevent their straightforward geometrical handling, and often imply a non-integer Hausdorff dimension.

The main goal of the thesis is to develop an alternative approach to the geometry of IFS fractals in the classical sense via bounding sets. The results are obtained with the objective of practical applicability.

The thesis thus revolves around the central problem of determining bounding sets to IFS fractals - and the convex hull in particular - emphasizing the fundamental role of such sets in their geometry. This emphasis is supported throughout the thesis, from real-life and theoretical applications to numerical algorithms crucially dependent on bounding.