On Intersecting IFS Fractals with Lines

TL;DR

This paper advances the understanding of how IFS fractals intersect with lines, providing verifiable conditions and algorithms for intersection detection, which benefits applications in graphics and antenna design.

Contribution

It introduces a verifiable condition for line intersection with hyperdense IFS fractals and offers a constructive algorithm for finding intersection points.

Findings

A verifiable condition guarantees intersection with lines through the convex hull.

An explicit formula for the invariant measure quantifies the intersection.

Infinite approximate intersections are guaranteed under certain conditions.

Abstract

IFS fractals - the attractors of Iterated Function Systems - have motivated plenty of research to date, partly due to their simplicity and applicability in various fields, such as the modeling of plants in computer graphics, and the design of fractal antennas. The statement and resolution of the Fractal-Line Intersection Problem is imperative for a more efficient treatment of certain applications. This paper intends to take further steps towards this resolution, building on the literature. For the broad class of hyperdense fractals, a verifiable condition guaranteeing intersection with any line passing through the convex hull of a planar IFS fractal is shown, in general R^d for hyperplanes. The condition also implies a constructive algorithm for finding the points of intersection. Under certain conditions, an infinite number of approximate intersections are guaranteed, if there is at least one. Quantification of the intersection is done via an explicit formula for the invariant measure of IFS.