Optimal Filling of Shapes

TL;DR

This paper explores optimal filling strategies for shapes, focusing on placing overlapping objects like n-balls and discs to maximize interior coverage, with theoretical insights and heuristics for polygons.

Contribution

It introduces the concept of filling as a spatial subdivision problem and provides methods for finding maximal fillings in various shapes, including polygons.

Findings

Solutions correspond to sets of maximal n-balls in n-dimensional space.

Heuristic methods are proposed for maximal disc placement in polygons.

Analysis of ideal distributions of discs as their number increases.

Abstract

We present filling as a type of spatial subdivision problem similar to covering and packing. Filling addresses the optimal placement of overlapping objects lying entirely inside an arbitrary shape so as to cover the most interior volume. In n-dimensional space, if the objects are polydisperse n-balls, we show that solutions correspond to sets of maximal n-balls. For polygons, we provide a heuristic for finding solutions of maximal discs. We consider the properties of ideal distributions of N discs as N approaches infinity. We note an analogy with energy landscapes.