Good Illumination of Minimum Range

TL;DR

This paper investigates optimizing the illumination range of point lights to ensure a point or line segment is 1-well illuminated, providing algorithms for minimal range and efficient querying.

Contribution

It introduces algorithms for minimizing light source range for 1-well illumination of points and segments, including preprocessing for fast range queries.

Findings

O(n^2) algorithm for segment illumination

Minimal illumination range computation for a point

Preprocessing enables O(log n) query time

Abstract

A point p is 1-well illuminated by a set F of n point lights if p lies in the interior of the convex hull of F. This concept corresponds to triangle-guarding or well-covering. In this paper we consider the illumination range of the light sources as a parameter to be optimized. First, we solve the problem of minimizing the light sources' illumination range to 1-well illuminate a given point p. We also compute a minimal set of light sources that 1-well illuminates p with minimum illumination range. Second, we solve the problem of minimizing the light sources' illumination range to 1-well illuminate all the points of a line segment with an O(n^2) algorithm. Finally, we give an O(n^2 log n) algorithm for preprocessing the data so that one can obtain the illumination range needed to 1-well illuminate a point of a line segment in O(log n) time. These results can be applied to solve problems of 1-well illuminating a trajectory by approaching it to a polygonal path.