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ORourkeCH5

# ORourkeCH5 - UMass Lowell Computer Science 91.504 Advanced...

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UMass Lowell Computer Science 91.504 Advanced Algorithms Computational Geometry Prof. Karen Daniels Spring, 2010 Voronoi Diagrams & Delaunay Triangulations O’Rourke: Chapter 5 de Berg et al.: Chapters 7, 9 (and a touch of Ch. 8)

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Voronoi Diagrams Applications Preview Definitions & Basic Properties Delaunay Triangulations Algorithms More on Applications Medial Axis Connection to Convex Hulls Connection to Arrangements
Applications Preview Nearest neighbors Minimum spanning tree Medial axis http://www.ics.uci.edu/~eppstein/gina/scot.drysdale.html

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Common 2D Computational Geometry Structures Voronoi Diagram Delaunay Triangulation Convex Hull New Point Combinatorial size in O(n) triangle whose circumcircle’s center is just outside triangle
Basics Voronoi region V(pi) is set of all points at least as close to pi as to any other site: } | | | :| { ) ( i j x p x p x p V j i i 2200 - - = j i j i i p p H p V = ) , ( ) ( Note:This definition is independent of dimension. source: O’Rourke

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Digression on Point/Line Duality Point in 2D plane has coordinates x and y (Non-vertical) line has slope and y -intercept One-to-one mapping of set of points to set of lines (and vice versa) is a type of duality mapping Example of one duality: source: de Berg et al. Ch 8 ) ( line is *) (denoted point of dual y x p x p y p p - = ) , ( * point is ) ( line of dual b m l b mx y l - = + = This duality preserves: - Incidence - Order (above/below)
Delaunay Triangulations: Properties http://www.cs.cornell.edu/Info/People/chew/Delaunay.html

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2D Delaunay Triangulations: Properties D1. D(P) is the straight-line dual of V(P) [by definition] D2. D(P) is a triangulation if no 4 points of P are cocircular: Every face is a triangle. D3. Each face of D(P) corresponds to a vertex of V(P) D4. Each edge of D(P) corresponds to an edge of V(P) D5. Each node of D(P) corresponds to a region of V(P) D6. The boundary of D(P) is the convex hull of the sites D7. The interior of each face of D(P) contains no sites source: O’Rourke D3-D5 define a duality mapping between features of the Delaunay Triangulation and Voronoi Diagram. P is the set of sites ( no 4 cocircular ).
2D Delaunay Triangulations: Properties of Voronoi Diagrams http://www.cs.cornell.edu/Info/People/chew/Delaunay.html

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2DVoronoi Diagram: Properties V1. Each Voronoi region V(pi) is convex V2. V(pi) is unbounded iff pi is on convex hull of point set V3. If v is a Voronoi vertex at junction of V(p1), V(p2), V(p3), then v is center of circle C(v) determined by p1, p2, and p3. V4. C(v) is circumcircle for Delaunay triangle for v V5. Interior(C(v)) contains no sites (proof by contradiction) V6. If pi is a nearest neighbor to pj, then (pi, pj) is an edge of D(P) V7. If there is some circle through pi and pj that contains no other sites, then (pi, pj) is an edge of D(P). [reverse holds too] source: O’Rourke
Back to Delaunay Triangulation: Another

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ORourkeCH5 - UMass Lowell Computer Science 91.504 Advanced...

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