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Unformatted text preview: Motivation Triangulating a polygon Triangulating a polygon Computational Geometry Lecture 4: Triangulating a polygon Computational Geometry Lecture 4: Triangulating a polygon Motivation Triangulating a polygon Visibility in polygons Triangulation Proof of the Art gallery theorem Polygons and visibility Two points in a simple polygon can see each other if their connecting line segment is in the polygon Computational Geometry Lecture 4: Triangulating a polygon Motivation Triangulating a polygon Visibility in polygons Triangulation Proof of the Art gallery theorem Art gallery problem Art Gallery Problem: How many cameras are needed to guard a given art gallery so that every point is seen? Computational Geometry Lecture 4: Triangulating a polygon Motivation Triangulating a polygon Visibility in polygons Triangulation Proof of the Art gallery theorem Art gallery problem In geometry terminology: How many points are needed in a simple polygon with n vertices so that every point in the polygon is seen? The optimization problem is computationally difficult Art Gallery Theorem: b n/ 3 c cameras are occasionally necessary but always sufficient Computational Geometry Lecture 4: Triangulating a polygon Motivation Triangulating a polygon Visibility in polygons Triangulation Proof of the Art gallery theorem Triangulation, diagonal Why are b n/ 3 c always enough? Assume polygon P is triangulated : a decomposition of P into disjoint triangles by a maximal set of nonintersecting diagonals Diagonal of P : open line segment that connects two vertices of P and fully lies in the interior of P Computational Geometry Lecture 4: Triangulating a polygon Motivation Triangulating a polygon Visibility in polygons Triangulation Proof of the Art gallery theorem A triangulation always exists Lemma: A simple polygon with n vertices can always be triangulated, and always have n 2 triangles Proof: Induction on n . If n = 3 , it is trivial Assume n > 3 . Consider the leftmost vertex v and its two neighbors u and w . Either uw is a diagonal (case 1), or part of the boundary of P is in 4 uvw (case 2) Choose the vertex t in 4 uvw farthest from the line through u and w , then vt must be a diagonal v v u u w w Computational Geometry Lecture 4: Triangulating a polygon Motivation Triangulating a polygon Visibility in polygons Triangulation Proof of the Art gallery theorem A triangulation always exists In case 1, uw cuts the polygon into a triangle and a simple polygon with n 1 vertices, and we apply induction In case 2, vt cuts the polygon into two simple polygons with m and n m + 2 vertices, 3 ≤ m ≤ n 1 , and we also apply induction By induction, the two polygons can be triangulated using m 2 and n m + 2 2 = n m triangles. So the original polygon is triangulated using m 2 + n m = n 2 triangles Computational Geometry Lecture 4: Triangulating a polygon Motivation Triangulating a polygon Visibility in polygons Triangulation Proof of the Art gallery theorem A 3coloring always exists...
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 Spring '11
 julianpleife
 Triangulating, eﬃcient algorithm Partitioning

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