Optimal Fillings - A new spatial subdivision problem related to packing and covering

TL;DR

This paper introduces the filling problem, a new spatial subdivision challenge related to packing and covering, focusing on optimal placement of overlapping objects within shapes to maximize interior coverage, with algorithms and theoretical insights.

Contribution

It defines the filling problem, relates it to maximal n-balls and the medial axis, and develops heuristics and genetic algorithms for polygon filling solutions.

Findings

Solutions correspond to sets of maximal n-balls.

The solution space reduces to the medial axis of the shape.

Heuristic and genetic algorithms effectively find maximal disc solutions.

Abstract

We present filling as a new type of spatial subdivision problem that is related 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 and the solution space can reduced to the medial axis of a shape. We examine the structure of the solution space in two dimensions. For the filling of polygons, we provide detailed descriptions of a heuristic and a genetic algorithm for finding solutions of maximal discs. We also consider the properties of ideal distributions of N discs in polygons as N approaches infinity.