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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)
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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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Page 1: 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)

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.

Page 2: 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)
http://upload.wikimedia.org/wikipedia/commons/2/20/Coloured_Voronoi_2D.svg
Page 3: 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)
Page 4: 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)
http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=Xg4BFsh3NkCgLM&tbnid=ClimIE0ID6Hv0M:&ved=0CAgQjRwwAA&url=http%3A%2F%2Fwww.dma.fi.upm.es%2Fmabellanas%2Ftfcs%2Ffvd%2Fvoronoi.html&ei=OowUUeWlDJG6yAHY64B4&psig=AFQjCNEES_jIB_vtCdMDiOt8xAe-D_OfDQ&ust=1360387514228086
Page 5: 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)
Page 6: 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)
Page 7: 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)
Page 8: 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)

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

Page 9: 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)

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.

Page 10: 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)

Uniform Triangle Mass CenterFractal Pattern

Page 11: 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)

Input Output

Page 12: 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)

Voronoi tessellation

based on world population density

Page 13: 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)

Voroni Map of United States based on state capital. It pretty much mirrors existing state boundaries; supports that the state capitals were located in the center of each state

http://www.google.com/url?sa=i&rct=j&q=voronoi+blogspot&source=images&cd=&cad=rja&docid=ZbCVKRDk3QA8fM&tbnid=BRG3DX5f6BolWM:&ved=0CAUQjRw&url=http%3A%2F%2Fgeographer-at-large.blogspot.com%2F2012%2F03%2Fmap-of-week-3-5-2012craigslist-meets.html&ei=PZQUUZyvHJKuqAHS2IDQDw&psig=AFQjCNHK3f0yXuqCbP1vy43iEVDzsdt7Zw&ust=1360389533919649
Page 14: 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)
Page 15: 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)

As long as I stay on the black lines, I will be as far away from the surrounding houses as I possibly can be.

http://www.google.com/url?sa=i&rct=j&q=voronoi+diagram&source=images&cd=&cad=rja&docid=ePN6-pNI1d9FnM&tbnid=JZDcEkY1Pl2A6M:&ved=0CAUQjRw&url=http%3A%2F%2Fpushypanda.blogspot.com%2F2011%2F02%2Fvoronoi-diagrams-and-applications.html&ei=35IUUazkD8jyqwHe5ICwAQ&bvm=bv.42080656,d.aWc&psig=AFQjCNHJDhse8iGK1jkxDdTxwfZI5wNKEg&ust=1360389170453142
Page 16: 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)

What if more houses?

Page 17: 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)
Page 18: 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)

Pizza Parlor Proximity

Page 19: 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)

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.

Page 20: 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)
Page 21: 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)
Page 22: 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)

Workers Needed to Service a Given Area

Page 23: 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)
Page 24: 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)
Page 25: 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)

Workers Needed to Service a Given Area

Page 26: 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)
Page 27: 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)
Page 28: 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)
Page 29: 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)
Page 30: 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)

Workers Needed to Service a Given Area

Page 31: 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)

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