Guarding orthogonal galleries with rectangular rooms

Consider an orthogonal art gallery partitioned into n rectangular rooms. If two rooms are adjacent, there is a door connecting them and a guard positioned at this door will see both rooms. In Czyzowicz et al. [(1994) Guarding rectangular art galleries. Discrete Appl. Math., 50, 149–157], it is shown that any rectangular gallery can be guarded with ⌈n/2⌉ guards. We prove that the same bound holds for L-shape polygons. We extend it to staircases and prove that an orthogonal staircase with n rooms and r reflex vertices can be guarded with ⌈(n+⌊ r/2⌋)/2⌉ guards. Then we prove an upper bound on the number of guards for arbitrary orthogonal polygon with orthogonal holes. This result improves the previous bound by Czyzowicz et al. [(1994) Guarding rectangular art galleries. Discrete Appl. Math., 50, 149–157] (even in the case of polygon without holes)

The three-dimensional art gallery problem and its solutions

This thesis addressed the three-dimensional Art Gallery Problem (3D-AGP), a version of the art gallery problem, which aims to determine the number of guards required to cover the interior of a pseudo-polyhedron as well as the placement of these guards. This study exclusively focused on the version of the 3D-AGP in which the art gallery is modelled by an orthogonal pseudo-polyhedron, instead of a pseudo-polyhedron. An orthogonal pseudopolyhedron provides a simple yet effective model for an art gallery because of the fact that most real-life buildings and art galleries are largely orthogonal in shape. Thus far, the existing solutions to the 3D-AGP employ mobile guards, in which each mobile guard is allowed to roam over an entire interior face or edge of a simple orthogonal polyhedron. In many realword applications including the monitoring an art gallery, mobile guards are not always adequate. For instance, surveillance cameras are usually installed at fixed locations. The guard placement method proposed in this thesis addresses such limitations. It uses fixedpoint guards inside an orthogonal pseudo-polyhedron. This formulation of the art gallery problem is closer to that of the classical art gallery problem. The use of fixed-point guards also makes our method applicable to wider application areas. Furthermore, unlike the existing solutions which are only applicable to simple orthogonal polyhedra, our solution applies to orthogonal pseudo-polyhedra, which is a super-class of simple orthogonal polyhedron. In this thesis, a general solution to the guard placement problem for 3D-AGP on any orthogonal pseudo-polyhedron has been presented. This method is the first solution known so far to fixed-point guard placement for orthogonal pseudo-polyhedron. Furthermore, it has been shown that the upper bound for the number of fixed-point guards required for covering any orthogonal polyhedron having n vertices is (n3/2), which is the lowest upper bound known so far for the number of fixed-point guards for any orthogonal polyhedron. This thesis also provides a new way to characterise the type of a vertex in any orthogonal pseudo-polyhedron and has conjectured a quantitative relationship between the numbers of vertices with different vertex configurations in any orthogonal pseudo-polyhedron. This conjecture, if proved to be true, will be useful for gaining insight into the structure of any orthogonal pseudo-polyhedron involved in many 3-dimensional computational geometrical problems. Finally the thesis has also described a new method for splitting orthogonal polygon iv using a polyline and a new method for splitting an orthogonal polyhedron using a polyplane. These algorithms are useful in applications such as metal fabrication

Guarding and Searching Polyhedra

Guarding and searching problems have been of fundamental interest since the early years of Computational Geometry. Both are well-developed areas of research and have been thoroughly studied in planar polygonal settings. In this thesis we tackle the Art Gallery Problem and the Searchlight Scheduling Problem in 3-dimensional polyhedral environments, putting special emphasis on edge guards and orthogonal polyhedra. We solve the Art Gallery Problem with reflex edge guards in orthogonal polyhedra having reflex edges in just two directions: generalizing a classic theorem by O'Rourke, we prove that r/2 + 1 reflex edge guards are sufficient and occasionally necessary, where r is the number of reflex edges. We also show how to compute guard locations in O(n log n) time. Then we investigate the Art Gallery Problem with mutually parallel edge guards in orthogonal polyhedra with e edges, showing that 11e/72 edge guards are always sufficient and can be found in linear time, improving upon the previous state of the art, which was e/6. We also give tight inequalities relating e with the number of reflex edges r, obtaining an upper bound on the guard number of 7r/12 + 1. We further study the Art Gallery Problem with edge guards in polyhedra having faces oriented in just four directions, obtaining a lower bound of e/6 - 1 edge guards and an upper bound of (e+r)/6 edge guards. All the previously mentioned results hold for polyhedra of any genus. Additionally, several guard types and guarding modes are discussed, namely open and closed edge guards, and orthogonal and non-orthogonal guarding. Next, we model the Searchlight Scheduling Problem, the problem of searching a given polyhedron by suitably turning some half-planes around their axes, in order to catch an evasive intruder. After discussing several generalizations of classic theorems, we study the problem of efficiently placing guards in a given polyhedron, in order to make it searchable. For general polyhedra, we give an upper bound of r^2 on the number of guards, which reduces to r for orthogonal polyhedra. Then we prove that it is strongly NP-hard to decide if a given polyhedron is entirely searchable by a given set of guards. We further prove that, even under the assumption that an orthogonal polyhedron is searchable, approximating the minimum search time within a small-enough constant factor to the optimum is still strongly NP-hard. Finally, we show that deciding if a specific region of an orthogonal polyhedron is searchable is strongly PSPACE-hard. By further improving our construction, we show that the same problem is strongly PSPACE-complete even for planar orthogonal polygons. Our last results are especially meaningful because no similar hardness theorems for 2-dimensional scenarios were previously known

Geometric Problems in Robot Exploration

Robots are increasingly utilized to perform tasks in today\u27s world. This has varied from vacuuming to building advanced structures. With robots being used for tasks such as these, new challenges are introduced. Problems that have been previously researched to be performed, either theoretically or implemented, need to be redesigned to be able to better handle these challenges. In this thesis, I will discuss multiple problems that have previously been researched and I have redesigned to be possible to be implemented by robots or that I have developed a new way for the robots to solve the problem. I focus on geometric areas and robots tasked with performing exploration in the area. Exploration is a task in which an unknown area is completely traversed. In this work, I have develop multiple algorithms to perform online tasks that can be implemented by robots. With robots performing exploration, a limited viewing range and communication range increase difficulty. These algorithms are focused on utilizing robots to perform Exploration and Concave Decomposition. The results from this thesis are such that the Exploration algorithm that given a fleet of robots k, the total area n can be explored in O(n/k) time with all agents having work O(n/k). The Concave decomposition task has multiple algorithms focusing on a different aspect. In the first, with an online algorithm, with r reflex points, I perform the decomposition generating r + 1 convex areas. The other two online algorithms focus on interior cut length, which previously has not been researched. In response, have developed an algorithm which maintains interior length relative to the perimeter

Exploring Topics of the Art Gallery Problem

Created in the 1970\u27s, the Art Gallery Problem seeks to answer the question of how many security guards are necessary to fully survey the floor plan of any building. These floor plans are modeled by polygons, with guards represented by points inside these shapes. Shortly after the creation of the problem, it was theorized that for guards whose positions were limited to the polygon\u27s vertices, the floor of n/3 guards are sufficient to watch any type of polygon, where n is the number of the polygon\u27s vertices. Two proofs accompanied this theorem, drawing from concepts of computational geometry and graph theory

Cerro Chepen and the Late Moche Collapse in the Jequetepeque Valley, North Coast of Peru

In this dissertation, I investigate the socio-political processes that led to the collapse of the Late Moche political communities located in the Lower Jequetepeque Valley, North Coast of Peru. During the Late Moche phase (AD 600 to 850), the human populations of this valley evidenced an interesting case of political fragmentation and internal conflict. The Moche collapse in the Jequetepeque Valley is approached from the perspective of one of the largest power centers of the region: the fortified site of Cerro Chepén. This site occupies the upper and eastern slopes of a hill, located in a relatively central position within this valley. The site is significant for presenting a sophisticated system of fortifications, and two clearly-defined occupation sectors (which I call Cerro Chepén Alto and Cerro Chepén Bajo). Of these two sectors, Cerro Chepén Alto distinguishes itself by occupying a dominant position on top of the hill, and by being surrounded by the most remarkable defenses. This sector houses up to nine monumental buildings. The four that occupy an advantageous, central position integrate architectural spaces of highland design. Three of these four central buildings were excavated to evaluate the hypothesis that they housed highland intruders. The assessment of the cultural identity of the buildings occupants was based on two aspects of the process of materialization of ideology that is common to most complex societies \u2013 namely, the design of monumental architecture and the style of prestige objects. The results of the architectural and fine ceramic analyses led me to conclude that the occupants of these structures came from sites located in the nearby highlands, possibly outliers related to the area of interaction of the ceremonial center of Marcahuamachuco. Paleoenvironmental data suggest that their arrival in the lower section of the valley coincided with a period of decreased rainfall in the highlands. The careful planning of the fortified redoubt suggests that the newcomers not only participated in the internal conflict that affected local communities, but possibly exacerbated existing tensions. The collapse would have arisen due to the tensions that are inherent to situations of internecine warfare.\u2

Single-crossing orthogonal axial lines in orthogonal rectangles

The axial map of a town is one of the key components of the space syntax method – a tool for analysing urban layout. It is derived by placing the longest and fewest lines, called axial lines, to cross the adjacencies between convex polygons in a convex map of a town. Previous research has shown that placing axial lines to cross the adjacencies between a collection of convex polygons is NP-complete, even when the convex polygons are restricted to rectangles and the axial lines to have orthogonal orientation. In this document, we show that placing orthogonal axial lines in orthogonal rectangles where the adjacencies between the rectangles are restricted to be crossed only once (ALPSC- OLOR) is NP-complete. As a result, we infer the single adjacency crossing version of the general axial line placement problem is NP-complete. The transformation of NPcompleteness of ALP-SC-OLOR is from vertex cover for biconnected planar graphs. A heuristic is then presented that gives a reasonable approximate solution to ALP-SC-OLOR based on a greedy method

Le Corbusier and the mysterious “résidence du président d’un collège”

[EN] At the very end of his travel to United States, Le Corbusier conceived and designed a modern villa that he lately inserted in the third volume of his Oeuvre Complete with the title ‘Résidence du président d’un college près Chicago’ and few words below describing it. He interpreted a simple request for suggestions by Joseph Brewer, the president of the Olivet College, Michigan, into an actual commission for a new house that responded to the kind of works he expected from his American admirers. He possibly designed it in a few hours’ time from Kalamazoo to Chicago but the autograph hand-drafted plans and bird’s-eye perspective view in the Oeuvre Complete congruently describe a well-thought project showing a number of affinities with his most celebrated European houses. The villa can be considered as an aware modular assemblage of parts that he had previously designed or even built, tied together by a long and suggestive promenade architecturale, to offer the “timid” American people a sort of full scale model to introduce them to his vision of modern life. By analyzing Le Corbusier’s sketches and conjecturing both dimensions and missing elements from previous designs, a threedimensional digital model has been elaborated to virtually visit the résidence and understand its fictive and educational value.Colonnese, F. (2016). Le Corbusier and the mysterious “résidence du président d’un collège”. En LE CORBUSIER. 50 AÑOS DESPUÉS. Editorial Universitat Politècnica de València. 422-440. https://doi.org/10.4995/LC2015.2015.774OCS42244