Abstract:
We consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves, with edge or mobile guards. Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly at the vertices, and their edges are convex arcs. We transform the problem of monitoring a piecewise-convex polygon to the problem of 2-dominating a properly defined triangulation graph with edges or diagonals, where 2-dominance requires that every triangle in the triangulation graph has at least two of its vertices in its 2-dominating set. We show that $\lfloor\frac{n+1}{3}\rfloor$ diagonal guards or $\lfloor\frac{2n+1}{5}\rfloor$ edge guards are always sufficient and sometimes necessary, in order to 2-dominate a triangulation graph. Furthermore, we show how to compute: a diagonal 2-dominating set of size $\lfloor\frac{n+1}{3}\rfloor$ in linear time, an edge 2-dominating set of size $\lfloor\frac{2n+1}{5}\rfloor$ in $O(n^2)$ time, and an edge 2-dominating set of size $\lfloor\frac{3n}{7}\rfloor$ in O(n) time. Based on the above-mentioned results, we prove that, for piecewise-convex polygons, we can compute: a mobile guard set of size $\lfloor\frac{n+1}{3}\rfloor$ in $O(n\log{}n)$ time, an edge guard set of size $\lfloor\frac{2n+1}{5}\rfloor$ in $O(n^2)$ time, and an edge guard set of size $\lfloor\frac{3n}{7}\rfloor$ in $O(n\log{}n)$ time. Finally, we show that $\lfloor\frac{n}{3}\rfloor$ mobile or $\lceil\frac{n}{3}\rceil$ edge guards are sometimes necessary. When restricting our attention to monotone piecewise-convex polygons, the bounds mentioned above drop: $\lceil\frac{n+1}{4}\rceil$ edge or mobile guards are always sufficient and sometimes necessary; such an edge or mobile guard set, of size at most $\lceil\frac{n+1}{4}\rceil$, can be computed in O(n) time.

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, $\lfloor\frac{4n}{7}\rfloor-1$ guards are sometimes necessary, and $\lfloor\frac{2n}{3}\rfloor$ guards are always sufficient. Moreover, an $O(n\log{}n)$ time and O(n) space algorithm is described that produces a vertex guarding set of size at most $\lfloor\frac{2n}{3}\rfloor$. When we allow point guards the afore-mentioned lower bound drops down to $\lfloor\frac{n}{2}\rfloor$. In the special case of monotone piecewise-convex polygons we can show that $\lfloor\frac{n}{2}\rfloor$ vertex guards are always sufficient and sometimes necessary; these bounds remain valid even if we allow point guards. In the case of piecewise-concave polygons, we show that $2n-4$ point guards are always sufficient and sometimes necessary, whereas it might not be possible to guard such polygons by vertex guards. We conclude with bounds for other types of curvilinear polygons and future work.

Abstract:
The Art Gallery Problem is one of the most well-known problems in Computational Geometry, with a rich history in the study of algorithms, complexity, and variants. Recently there has been a surge in experimental work on the problem. In this survey, we describe this work, show the chronology of developments, and compare current algorithms, including two unpublished versions, in an exhaustive experiment. Furthermore, we show what core algorithmic ingredients have led to recent successes.

Abstract:
We introduce the notion of a normal gallery, a gallery in which any configuration of guards that visually covers the walls covers the entire gallery. We show that any star gallery is normal and any gallery with at most two reflex corners is normal. A polynomial time algorithm is provided deciding if, for a given polygon and a finite set of positions, there exists a configuration of guards in some of these positions that visually covers the walls but not the entire gallery.

Abstract:
We consider guarding classes of simple polygons using mobile guards (polygon edges and diagonals) under the constraint that no two guards may see each other. In contrast to most other art gallery problems, existence is the primary question: does a specific type of polygon admit some guard set? Types include simple polygons and the subclasses of orthogonal, monotone, and starshaped polygons. Additionally, guards may either exclude or include the endpoints (so-called open and closed guards). We provide a nearly complete set of answers to existence questions of open and closed edge, diagonal, and mobile guards in simple, orthogonal, monotone, and starshaped polygons, with some surprising results. For instance, every monotone or starshaped polygon can be guarded using hidden open mobile (edge or diagonal) guards, but not necessarily with hidden open edge or hidden open diagonal guards.

Abstract:
This article presents a generalization of the standard art gallery problem to the case where the sides of the gallery are continuous curves which are limits of polygonal arcs. The allowable limiting processes for such generalized art galleries are defined. We construct an art gallery in which one side is the Koch fractal and the other sides are three sides of a rectangle. The appropriate measure of coverage by guards is not the total number of guards but, rather, the guards-to-side ratio. We compute this ratio for the cases of shallow and deep versions of the Koch fractal art gallery.

Abstract:
This paper focuses on a variation of the Art Gallery problem that considers open edge guards and open mobile guards. A mobile guard can be placed on edges and diagonals of a polygon, and the "open" prefix means that the endpoints of such edge or diagonal are not taken into account for visibility purposes. This paper studies the number of guards that are sufficient and sometimes necessary to guard some classes of simple polygons for both open edge and open mobile guards. This problem is also considered for planar triangulation graphs using open edge guards.

Abstract:
This paper is focused on the public presentation of self through virtual art galleries, singling out the field of photography. Photography has always been disputed as being part of the highbrow arts because of its popular character. Today, anyone who owes a photo camera can experience photography as art, without a rigorous training. Everybody is able to expose the photos freely to a large number of people, on the Internet. Consequently, the Internet opens up a virtual space, in which photo artists and amateurs can promote their works and exhibit them in a personal online gallery, which represents their place in the virtual vastness. Therefore, my research approaches the matter of the virtual gallery as an identitary place, being focused on finding out why artists choose to exhibit in virtual galleries. I asked myself what are the new functions of the virtual art galleries?Are they understood as online markers that distinguish the owners in these virtual environments? In other words, are these personal galleries a way of expressing online identities?

Abstract:
We consider a generalization of the classical Art Gallery Problem, where instead of a light source, the guards, called $k$-transmitters, model a wireless device with a signal that can pass through at most $k$ walls. We show it is NP-hard to compute a minimum cover of point 2-transmitters, point $k$-transmitters, and edge 2-transmitters in a simple polygon. The point 2-transmitter result extends to orthogonal polygons. In addition, we give necessity and sufficiency results for the number of edge 2-transmitters in general, monotone, orthogonal monotone, and orthogonal polygons.