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Voronoi Diagrams 1. Given a set of points S in the plane, which are the Voronoi sites. 2. Each site s has a Voronoi cell, also called a Dirichlet cell, V(s) consisting of all points closer to s than to any other site. 3. The segments of the Voronoi diagram are all the points in the plane that are equidistant to the two nearest sites. 4. The Voronoi nodes are the points equidistant to three (or more) sites.

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Voronoi Diagrams. Given a set of points S in the plane, which are the Voronoi sites. Each site s has a Voronoi cell, also called a Dirichlet cell, V(s) consisting of all points closer to s than to any other site. - PowerPoint PPT Presentation

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Voronoi Diagrams

1. Given a set of points S in the plane, which are the Voronoi sites.

2. Each site s has a Voronoi cell, also called a Dirichlet cell, V(s) consisting of all points closer to s than to any other site.

3. The segments of the Voronoi diagram are all the points in the plane that are equidistant to the two nearest sites.

4. The Voronoi nodes are the points equidistant to three (or more) sites.

Where should the shops be built?

Ten shops in a flat city and their Voronoi cells

Based on this model, what is the assumption regarding how customers are choosing which shop to go to?

1. Distance is only consideration 2. Euclidean Distance is used. 3. Customers travel by most efficient

path Point A to Point B

Where should the shops be built?Ten shops in a flat city and their Voronoi cells

Based on this model, what is the assumption regarding how customers are choosing which shop to go to and the path they take? 1. Distance is only consideration 2. Manhattan Distance is used. 3. Customers only go to the shops

by a vehicle and the traffic paths are parallel to the and axes

Note: Voronoi cells depend significantly on the metric used.

Uniform Triangle Mass CenterFractal Pattern

Input Output

Voronoi tessellation

based on world population density

What if more houses?

Pizza Parlor Proximity

You are the owner….1. …of five pizzerias in the town of

Squaresville. 2. To ensure minimal delivery times,

you devise a system in which customers call a central phone number and are then transferred to the pizzeria that is closest to them.

3. The map of Squaresville shows the position of the five pizzerias.

4. You need to divide the town into five regions so that customers order their pizza from the closest pizzeria.

Workers Needed to Service a Given Area

Workers Needed to Service a Given Area

Workers Needed to Service a Given Area