Voronoi2D - Lecture 7: Voronoi Diagrams Presented by Allen...

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Lecture 7: Voronoi Diagrams Presented by Allen Miu 6.838 Computational Geometry September 27, 2001
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Post Office: What is the area of service? q q : free point e e : Voronoi edge v v : Voronoi vertex p i p i : site points
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Definition of Voronoi Diagram Let P be a set of n distinct points (sites) in the plane. The Voronoi diagram of P is the subdivision of the plane into n cells, one for each site. • A point q lies in the cell corresponding to a site p i P iff Euclidean_Distance ( q , p i ) < Euclidean_distance ( q , p j ), for each p i P, j i .
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Voronoi Diagram Example: 1 site
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Two sites form a perpendicular bisector Voronoi Diagram is a line that extends infinitely in both directions, and the two half planes on either side.
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Collinear sites form a series of parallel lines
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Non-collinear sites form Voronoi half lines that meet at a vertex A Voronoi vertex is the center of an empty circle touching 3 or more sites. v Half lines A vertex has degree 3
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Voronoi Cells and Segments v
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Voronoi Cells and Segments v Unbounded Cell Bounded Cell Segment
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Who wants to be a Millionaire? v Which of the following is true for 2-D Voronoi diagrams? Four or more non-collinear sites are… 1. sufficient to create a bounded cell 2. necessary to create a bounded cell 3. 1 and 2 4. none of above
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Who wants to be a Millionaire? v Which of the following is true for 2-D Voronoi diagrams? Four or more non-collinear sites are… 1. sufficient to create a bounded cell 1. necessary to create a bounded cell 1. 1 and 2 2. none of above
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Degenerate Case: no bounded cells! v
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Summary of Voronoi Properties A point q lies on a Voronoi edge between sites p i and p j iff the largest empty circle centered at q touches only p i and p j A Voronoi edge is a subset of locus of points equidistant from p i and p j e e : Voronoi edge v v : Voronoi vertex p i p i : site points
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Summary of Voronoi Properties A point q is a vertex iff the largest empty circle centered at q touches at least 3 sites A Voronoi vertex is an intersection of 3 more segments, each equidistant from a pair of sites e e : Voronoi edge v v : Voronoi vertex p i p i : site points
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Outline Definitions and Examples Properties of Voronoi diagrams Complexity of Voronoi diagrams Constructing Voronoi diagrams Intuitions Data Structures Algorithm Running Time Analysis Demo Duality and degenerate cases
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Voronoi diagrams have linear complexity {| v | , | e | = O( n )} Intuition: Not all bisectors are Voronoi edges! e e : Voronoi edge p i p i : site points
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Voronoi diagrams have linear complexity {| v | , | e | = O( n )} Claim: For n 3, | v | 2 n - 5 and | e | 3 n - 6 Proof: (Easy Case) Collinear sites | v | = 0, | e | = n – 1
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Voronoi diagrams have linear complexity {| v | , | e | = O( n )} Claim: For n 3, | v | 2 n - 5 and | e | 3 n - 6 Proof: (General Case) Euler’s Formula: for connected, planar graphs, | v | | e | + f = 2 Where: | v | is the number of vertices | e | is the number of edges f is the number of faces
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Voronoi2D - Lecture 7: Voronoi Diagrams Presented by Allen...

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