## 1 Introduction  
A large and growing body of literature about computational geometry has explored the art gallery problem. The main goal in this problem is to find the minimum number of point guards inside a polygon (\(P\)) such that the set of guards can see the whole \(P\). Each guard can see its surroundings in 360 degrees and up to infinity (if there are no obstacles). The number of guards that suffices and is sometimes necessary for any arbitrary polygon with \(n\) vertices is \(\lfloor n/3\rfloor\)[1]. The main goal in our study is to find the minimum set of angle guards by which the geometry of the polygon can be defined through two operations, \(AND(.)\) and \(OR(+)\). An _angle guard_\(g\) views an infinite wedge of the plane (by going through the involved obstacles) and can be defined as a Boolean predicate, \(B_{g}(p)\), which is _True_ for a given point, \(p\in P\), if \(p\) is inside the view region of \(g\), otherwise it is False. Given a polygon \(P\), the aim is to place a set of angle guards on \(P\) in such a way that the monotone Boolean formula \(F_{P}(p)\) is _True_, if and only if \(p\) is inside \(P\) or on the boundary of polygon \(P\) (\(p\in\partial P\)), otherwise it is _False_:  
\[F_{P}(p)=\left\{\begin{array}{ll}True&\text{if }p\in P\text{ or }p\in\partial P\\ False&\text{otherwise}\end{array}\right.\]There are several applications in machine vision, image processing and gamification which require the range of a shape to be determined. Various algorithms for these problems have been presented that are based on pixels, area, vertices and so on. These methods require a lot of processing time in problems that require fast processing (such as computer games and graphics). Even these methods force users to provide more powerful hardware to run a machine vision system or computer game. The method discussed in this paper is related to a problem that uses only a few points, angles and two logical operators to determine the range of a shape. Using this method, various applications in machine vision and computer games are performed much faster and with a lower number of operations. An angle guard vertex placement is considered as _natural_ if all the guards of \(P\) have the same view of their corresponding vertices [2]. As [PERSON] et al. stated, a polygon \(P\) can be demonstrated in a way that a natural angle guard vertex placement cannot fully distinguish between points on the inside and outside of \(P\) which implies that _Steiner_-point guards are sometimes necessary [2]. According to Figure 1a, even the placement of a natural angle at every vertex of the pentagon is not able to distinguish between the points \(x\) and \(y\) and at least one unnatural guard is needed to define the polygon (Figure 1b). Consequently, the polygon is defined by \(F=A.B.D\).  
A variety of cases of the problem is present in which the location and angle of view of the guards are different. We focus on a type of the problem that all the guards are placed on the vertices of \(P\). It was a conjecture [3] that in the worst case, \(n-2\) guards are needed to describe an \(n\)-gon. In this paper, we present an algorithm to generate a polygon for a given \(n\) which needs at least \(n-2\) natural vertex guards. We do not prove [PERSON]'s conjecture and this problem is still open. However, by using our algorithm, we create an \(n\)-gon for any given \(n\), which needs exactly \(n-2\) natural vertex guards.  
In the next section, the sculpture garden problem is introduced and some applications are mentioned. Section 3 provides the main problem and presents an algorithm for generating the \(n\)-gon which needs exactly \(n-2\) natural vertex guards to be defined. Finally, Section 4 presents the findings of the study and also some suggestions for further research.