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

This preview shows pages 1–20. Sign up to view the full content.

Lecture 7: Voronoi Diagrams Presented by Allen Miu 6.838 Computational Geometry September 27, 2001

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Demo http://www.diku.dk/students/duff/Fortune/ http://www.msi.umn.edu/~schaudt/voronoi/ voronoi.html
Voronoi Diagram Example: 1 site

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
Collinear sites form a series of parallel lines

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
Voronoi Cells and Segments v

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Voronoi Cells and Segments v Unbounded Cell Bounded Cell Segment
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
Degenerate Case: no bounded cells! v

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 79

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

This preview shows document pages 1 - 20. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online