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CG_PolygonTriangulation - Motivation Triangulating a...

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Motivation Triangulating a polygon Triangulating a polygon Computational Geometry Lecture 4: Triangulating a polygon Computational Geometry Lecture 4: Triangulating a polygon
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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
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Motivation Triangulating a polygon Visibility in polygons Triangulation Proof of the Art gallery theorem Art gallery problem Art Galley 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
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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 Galley Theorem: n/ 3 cameras are occasionally necessary but always sufficient Computational Geometry Lecture 4: Triangulating a polygon
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Motivation Triangulating a polygon Visibility in polygons Triangulation Proof of the Art gallery theorem Triangulation, diagonal Why are n/ 3 always enough? Assume polygon P is triangulated : a decomposition of P into disjoint triangles by a maximal set of non-intersecting 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
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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 uvw (case 2) Choose the vertex t in 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
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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
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Motivation Triangulating a polygon Visibility in polygons Triangulation Proof of the Art gallery theorem A 3-coloring always exists Observe: the dual of a triangulated simple polygon is a tree Lemma:
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