Art Gallery Theorem

Art Gallery Theorem Computational Geometry, WS 2006/07 Lecture 8, Part 1 Prof. Dr. Thomas Ottmann Algorithmen & Datenstrukturen, Institut für Informatik Fakultät für Angewandte Wissenschaften Albert-Ludwigs-Universität Freiburg Agenda • • Motivation: Guarding art galleries Art gallery theorem for simple polygons • • Partitioning of polygons into monotone pieces Triangulation of y-monotone polygons Computational Geometry, WS 2006/07 Prof. Dr. Thomas Ottmann 2 Guarding art galleries Problem Definition Imagine an art gallery room whose floor plan can be modeled by a polygon of n vertices. Victor Klee asked (1973): How many stationary guards are needed to guard the room? Computational Geometry, WS 2006/07 Prof. Dr. Thomas Ottmann 3 Guarding art galleries The gallery is represented by a simple polygon  A guard is represented by a point within the polygon  Guards have a viewport of 360°, and of course cannot see through a wall  A polygon is completely guarded, if every point within the polygon is guarded by at least one of the watchmen  Visibility polygon: The visibility polygon of a polygon P is defined by the set of all points that are visible from a base point p. Demo Computational Geometry, WS 2006/07 Prof. Dr. Thomas Ottmann 4 Guarding art galleries Even if two polygons have the same number of vertices, one may be easier to guard than the other. We are NOT interested in the minimum number of guards for a specific polygon, but rather want to determine the number of guards that suffice for an arbitrary polygon with n vertices. Computational Geometry, WS 2006/07 Prof. Dr. Thomas Ottmann 5 Guarding art galleries If the polygon is complex, it is not obvious to see how many gurads are needed. Idea: Divide the polygon into pieces that are easy to guard Computational Geometry, WS 2006/07 Prof. Dr. Thomas Ottmann 6 Guarding a triangulated polygon Computational Geometry, WS 2006/07 Prof. Dr. Thomas Ottmann 7 Triangulation of simple polygons Does every simple polygon admit a triangulation? If yes, what is the number of triangles? Does any triangulation of a polygon P lead to the same number of triangles? Computational Geometry, WS 2006/07 Prof. Dr. Thomas Ottmann 8 Theorem Theorem: Every simple polygon admits a triangulation. Proof: By induction on n. Let n>3, and assume theorem is true for all m3, and assume theorem is true for all m Every triangle has all 3 colors. Hence every triangle is watched. Hence the entire polygon is watched. Computational Geometry, WS 2006/07 Prof. Dr. Thomas Ottmann 14 Example Computational Geometry, WS 2006/07 Prof. Dr. Thomas Ottmann 15 3- coloring Theorem: The triangulation graph of a polygon P is 3-colorable. Proof: Induction on n. Clearly, a triangle can be 3-colored. Let n>3, and assume theorem is true for all m