How to Sketch a Voronoi Diagram with Thiessen Polygons

Voronoi Diagram Thiessen Polygons
United States Airports: Voronoi Diagram and Thiessen Polygons

You’re talking on your mobile phone…

Your network provider has a series of cell towers. When you make a call, which tower should your mobile phone connect to?

If all things are equal, the closest tower!

But how do you know what is the closest tower? It turns out that the answer lies in your Voronoi diagram.

The Voronoi Diagram and Thiessen Polygons

Let’s plot out a set of control points on a map. Each point has a specific color.

Random Points Voronoi Diagram

For every other pixel on the image, the Thiessen polygon takes the color of the nearest control point. In other words, when you take a point in any given Thiessen polygon, it indicates that it’s closer to that generating point than to any other.

Voronoi Diagram Thiessen Polygons Result

For example, if you walk in this Thiessen polygon of the Voronoi diagram, your mobile phone should connect to this tower highlighted in red. This generating point is closest because the point is in this Thiessen polygon.

Voronoi Diagram Example 1

Likewise, when you walk over here, this is now the closest tower that it would likely connect to.

Voronoi Diagram Example 2

As a result, this illustrates a cellular coverage map. When you add a point anywhere on the map (entirely up to you), that is which cell tower their call will be routed to.

Pretty neat application of Voronoi Diagrams, don’t you think? Now, let’s create our own.

Let’s Make Our Own Voronoi Diagram

Now that you have an intuitive understanding of how a Voronoi diagram can be used, let’s make our very own.

Here is the spatial distribution of airports in the United States:

United States Airports

In ArcGIS, select the  Analysis > Proximity > Create Thiessen Polygons  tool. After running the tool, here is the resulting Voronoi Diagram:

Voronoi Diagram Thiessen Polygons
United States Airports: Voronoi Diagram and Thiessen Polygons

You can now see which airports are the most remote in the United States by the size of the Thiessen polygon.

Here is the World Airports Voronoi displaying how the airport on Easter Island (Mataveri Airport at 27°S, 109°W) is the most remote with the nearest airport about 2,600km away from it.

Now, It’s Your Turn

Voronoi diagrams are not just pretty pictures. As noted, they help understand proximity and distance of features.

Georgy Voronyi is the creator of the Voronoi Diagram.

Consequently, his diagram is now being used in study areas like biology, networking and geo-science.

What are some interesting applications of Voronoi Diagrams that you know?

1 Comment on How to Sketch a Voronoi Diagram with Thiessen Polygons

  1. I understand why use it for cell towers. But why airports? If I would travel by car or public transport to “nearest” (fastest reached) airports wouldn´t quality and quantity of roads be very important parameter?

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