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The three-dimensional art gallery problem and its solutions

By Jefri Marzal

Abstract

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

Year: 2012
OAI identifier: oai:researchrepository.murdoch.edu.au:13508
Provided by: Research Repository

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Citations

  1. (1975). A combinatorial theorem in plane geometry,"
  2. (2003). A genetic algorithm for minimum tetrahedralization of a convex polyhedron,"
  3. (1995). A lagrangian-based heuristic for large-scale set covering problems,"
  4. (1975). A polyhedron representation for computer vision,"
  5. (1978). A short proof of Chavatal’s watchman theorem,"
  6. (1998). Advanced Algebra.
  7. Algorithm for Set Covering Problems,"
  8. (1991). Algorithmic aspects of alternating sum of volumes. Part 1: Data structure and difference operation,"
  9. (2009). An algorithm for splitting arbitrary polygon," presented at the
  10. (1983). An alternate proof of the rectilinear art gallery theorem,"
  11. (1981). An efficient algorithm for decomposing a polygon into starshaped polygons,"
  12. (1995). An efficient algorithm for guard placement in polygons with holes,"
  13. (2006). and N.Sitchinava, "Guard placement for Wireless Localization,"
  14. (2003). Angle Counts for Isothetic Polygons and Polyhedra,"
  15. (2008). Approximate convex deocmposition of polyhedra and its applications,"
  16. (2003). Approximation algorithm.
  17. (1987). Approximation algorithms for art gallery problems,"
  18. (2000). Art gallery and Illumination Problems.
  19. (2000). Art gallery problem with guards whose range vision is 180o,"
  20. (1987). Art Gallery Theorems and Algorithms:
  21. (1986). Computational complexity of art gallery problems,"
  22. (1998). Computational Geometry
  23. (2002). Computer-Aided Geometric Design:
  24. (2002). Connected component labeling based on the EVM Model,"
  25. (2010). Constraint-based modeling of Minimum Set Covering: Application to Species Differentation," Master, Departamento de Informática, Universidade Nova de
  26. (1991). Construcitve Non-Regularized Geometry,"
  27. (1992). Convex decomposition of polyhedra and robustness,"
  28. (1993). Discrete Images, Objects, and Functions in Z n.
  29. (1995). Domain extension of isothetic polyhedra with minimal CSG representation,"
  30. (1998). Edge guards in rectilinear polygons," computational Geometry:
  31. (1985). Edge-Based Data Structure for Solid Modlling in Curve-Surface Environments,"
  32. (1996). Efficient decomposition of polygons into L-shapes with application to VLSI layouts,"
  33. (1990). Elementary data structure with Pascal:
  34. (2008). Estimating the maximum hidden vertex set in polygons,"
  35. (2010). Exact Exponential Algorithms.
  36. (2011). Face guards for art galleries," presented at the XIV Spanish Meeting on Computational Geometry,
  37. (1988). Fast algorithm for polygon decomposition,"
  38. (1995). Finding the Largest Rectangle in Several Classes of Polygons," Harvard University,
  39. (1983). Galleries need fewer mobile guards: a variation on Chavatal’s theorem,"
  40. (1996). Generalized guarding and partitioning for rectilinear polygons,"
  41. (2008). Guarding curvilinear art galleries with edge or mobile guards,"
  42. (1997). Guarding polyhedral terrains,"
  43. (2005). How to place efficiently guards and paintings in an art gallery,"
  44. (1993). Illuminating rectangles and triangles in the plane,"
  45. (1990). Introduction to Algorithms. Cambridge:
  46. (2008). K-vertex guarding simple polygon,"
  47. (2009). Linear Algebra Theory and Applications.
  48. (2007). Locating guards for visibility coverage of polygons,"
  49. (2000). Logical analyis of numerical data,"
  50. (2009). Metaherustic Approaches for the Minimum Vertex Guard Problem,"
  51. (2000). Minimal simplicial dissections and triangulations of convex 3-polytopes,"
  52. (2004). Minimal tetrahedralizations of a class of polyhedra,"
  53. (1990). Minimum rectangular paritition problem for simple rectilinear polygons,"
  54. (2008). Minimum Rectilinear Partitioning.
  55. Multiple Polyline to Polygon Matching,"
  56. (1989). On the difficulty of tetrahedralizing 3-dimensional non-convex polyhedra,"
  57. (1988). On the representation and manipulation of regid solids,"
  58. (2006). On visibility problems in the plane - solving minimum vertex guard problems by successive approximation,"
  59. (1992). Optimal binary space partitions for orthogonal objects,"
  60. (1996). Optimum partitioning of rectilinear layouts,"
  61. Optimum partitioning problem for rectilinear VLSI layout,"
  62. (1997). Orthogonal polyhedra as geometric bounds in constructive solid geometry," Solid Modeling,
  63. (1999). Orthogonal Polyhedra: Representation and Computation,"
  64. (1986). Partitioning a polygonal region into trapezoids,"
  65. (2002). Partitioning orthgonal polygons into fat rectangles in polinomials time,"
  66. (1985). Pattern recognition and geometrical complexity,"
  67. Polygon decomposition,"
  68. (1974). Polyhedron Models:
  69. (1985). Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi Diagrams,"
  70. (2009). Problem solving and programming concept, eighth ed.
  71. (2005). Real-time collision detection.
  72. (1992). Recent results in art galleries,"
  73. (2007). Reconstructing orthogonal polyhedra from putative vertex sets," technical reports,
  74. (2008). Reconstruction of orthogonal polyhedra,"
  75. (1991). Rectangular partition is polynomial in two dimensions but NP-complete in three,"
  76. (1984). Rectilinear computational geometry,"
  77. (1973). Regular polytopes.
  78. (1998). Solving point and plane vs orthogonal polyhedra using the extreme vertices model (EVM)," presented at the
  79. (1996). Splitting a complex of convex polytopes in any dimension,"
  80. (1984). Stationing guards in rectilinear art galleries,"
  81. (2010). Steinitz theorems for orthogonal polyhedra,"
  82. (1911). Theorem on the simple finite polygon and polyhedron,"
  83. (1998). Theory of Linear and Integer Programming.
  84. (1992). Three-coloring the vertices of a triangulated simple polygon,"
  85. (1983). Traditional galleries require fewer watchmen,"
  86. (1991). Triangulating a simple polygon in linear time,"
  87. (1978). Triangulating a simple polygon,"
  88. (1995). Two NP-hard art-gallery probles for orthgonal polygons,"
  89. (1988). Uniqueness of orthogonal connect-the-dots,"
  90. (2005). Use of GIS for planning visual surveillace installations,"
  91. (2004). When can a graph form an orthgonal polyhedron?," presented at the CCCG
  92. (1990). Zone Theorem and Polyhedral Decompositions," Purdue University8-3-1990