Guarding curvilinear art galleries with vertex or point guards
Menelaos I. Karavelas, Elias P. Tsigaridas

TL;DR
This paper extends the art gallery problem to polygons with curvilinear walls, providing bounds and algorithms for guarding piecewise-convex and piecewise-concave polygons with vertex and point guards.
Contribution
It introduces bounds and efficient algorithms for guarding curvilinear polygons, a novel extension of the classical art gallery problem.
Findings
Vertex guards are sometimes necessary in piecewise-convex polygons.
An $O(n ext{log}n)$ algorithm produces a guard set of size at most $rac{2n}{3}$.
Point guards can reduce the number needed compared to vertex guards.
Abstract
One of the earliest and most well known problems in computational geometry is the so-called art gallery problem. The goal is to compute the minimum possible number guards placed on the vertices of a simple polygon in such a way that they cover the interior of the polygon. In this paper we consider the problem of guarding an art gallery which is modeled as a polygon with curvilinear walls. Our main focus is on polygons the edges of which are convex arcs pointing towards the exterior or interior of the polygon (but not both), named piecewise-convex and piecewise-concave polygons. We prove that, in the case of piecewise-convex polygons, if we only allow vertex guards, guards are sometimes necessary, and guards are always sufficient. Moreover, an time and O(n) space algorithm is described that produces a vertex…
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