The Visibility Center of a Simple Polygon

TL;DR

This paper introduces the concept of the visibility center in simple polygons, providing efficient algorithms to compute it for multiple points and all points within the polygon, with applications to geodesic center problems.

Contribution

It defines the visibility center of a point set inside a polygon and presents efficient algorithms to compute it, reducing the problem to geodesic center computations of half-polygons.

Findings

Algorithm for computing the visibility center of m points in O((n+m) log(n+m)) time.

Algorithm for finding the visibility center of all points in O(n log n) time.

Reduction of the visibility center problem to geodesic center of half-polygons with O((n+k) log(n+k)) complexity.

Abstract

We introduce the \emph{visibility center} of a set of points inside a polygon -- a point $c_V$ such that the maximum geodesic distance from $c_V$ to see any point in the set is minimized. For a simple polygon of $n$ vertices and a set of $m$ points inside it, we give an $O((n+m) \log {(n+m)})$ time algorithm to find the visibility center. We find the visibility center of \emph{all} points in a simple polygon in $O(n \log n)$ time. Our algorithm reduces the visibility center problem to the problem of finding the geodesic center of a set of half-polygons inside a polygon, which is of independent interest. We give an $O((n+k) \log (n+k))$ time algorithm for this problem, where $k$ is the number of half-polygons.