The visible perimeter of an arrangement of disks

TL;DR

This paper investigates how to order opaque disks to maximize their visible perimeter from above, providing bounds based on the arrangement of their centers and addressing related open questions.

Contribution

It establishes bounds on the maximum visible perimeter for arrangements of disks based on the density and distribution of their centers, partially answering open problems.

Findings

For dense point sets, the visible perimeter can be Omega(n^2/3).

On a small grid, the bound cannot be improved beyond Omega(n^2/3).

When centers are densely packed with small maximum distance, the visible perimeter is O(n^3/4).

Abstract

Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter---the total length of all pieces of their boundaries visible from above. We prove that if the centers of the disks form a dense point set, i.e., the ratio of their maximum to their minimum distance is O(n^1/2), then there is a stacking order for which the visible perimeter is Omega(n^2/3). We also show that this bound cannot be improved in the case of a sufficiently small n^1/2 by n^1/2 uniform grid. On the other hand, if the set of centers is dense and the maximum distance between them is small, then the visible perimeter is O(n^3/4) with respect to any stacking order. This latter bound cannot be improved either. Finally, we address the case where no more than c disks can have a point in common. These results partially answer some questions of Cabello, Haverkort, van Kreveld, and Speckmann.