On the Exact Convex Hull of IFS Fractals

TL;DR

This paper investigates the exact convex hull of IFS fractals, providing new methods for precise computation and analyzing conditions for solvability based on IFS parameters.

Contribution

It introduces two novel methods for exactly determining the convex hull of IFS fractals, enhancing practical applicability and understanding of extremal point finiteness.

Findings

Finiteness of extremal points depends on IFS parameters.

Two new methods for exact convex hull computation are proposed.

Focal periodicity is key to identifying extremal points.

Abstract

The problem of finding the convex hull of an IFS fractal is relevant in both theoretical and computational settings. Various methods exist that approximate it, but our aim is its exact determination. The finiteness of extremal points is examined a priori from the IFS parameters, revealing some cases when the convex hull problem is solvable. Former results are detailed from the literature, and two new methods are introduced and crystallized for practical applicability -- one more general, the other more efficient. Focal periodicity in the address of extremal points emerges as the central idea.