The Smallest Enclosing Disc of an Affine Ifs Fractal

Abstract

We present a novel approach to approximate the smallest disc to enclose an affine IFS attractor at any accuracy. The approach is efficient and characterized by very low memory requirements. It is based on a concept of spanning points we introduce to describe the extent of an IFS attractor. We give the formal definition of these points and show how to construct them efficiently in practice. A proof that the smallest disc to enclose the spanning points includes the smallest enclosing disc of the IFS attractor is given. We also show that the former disc approximates the latter with an error depending on parameters relevant to the construction of the points. The main benefit of using the spanning points is that their number is usually small even for high approximation accuracies. Thus, we can apply with success one of the known, smallest enclosing disc algorithms to the spanning points. The enclosed experimental results obtained for popular IFS fractals confirm the theoretical assertions of the paper, by showing that our method performs well in practice.

Approximate convex hull of affine iterated function system attractors

“…More recently, Martyn [11] presented a novel approach to approximate the smallest disc to enclose an affine IFS attractor at any accuracy. The method is based on a concept of spanning points he introduced to describe the extent of an IFS attractor.…”