This preview shows pages 1–9. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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...
View
Full
Document
 Spring '11
 julianpleife

Click to edit the document details