Also known as 'Voronoi networks^{1}' and 'Delaunay triangulations^{2}', Thiessen polygons were independently discovered in several fields of study, including climatology and geography. They are named after a climatologist who used them to perform a transformation from point climate stations to watersheds.

Thiessen polygons can be used to describe the area of influence of a point in a set of points. If you take a set of points and connect each point to its nearest neighbour, you have what's called a triangulated irregular network (TIN). If you bisect^{3} each connecting line segment perpendicularly^{4} and create closed polygons with the perpendicular bisectors, the result will be a set of Thiessen polygons. The area contained in each polygon is closer to the point on which the polygon is based than to any other point in the dataset.

**What Are They Used For?**

If you have a set of features or events that are represented as points and you wish to determine the area of influence of each individual event or feature, you can create a set of Thiessen polygons based on the points.

**An Example**

Suppose you work for a city and you want to know the catchment area for each library in your city - ie, the area that the library users come from. If you have a library with a very large catchment area, you might want to consider building a new library.

There are several different ways you can calculate this, depending on how much you know about each library and how much time and money you have.

You could simply say that the catchment area for each library is a circle with a radius of 5km, centred on each library. However, if you have two libraries closer than 5km to each other, you have overlap, which may be accurate, but doesn't give you an exclusive zone for each library. If the libraries are spread out, you will have areas that are not in the catchment zone for any library - in fact, if you can't overlap the circles, there will always be areas that are not in any catchment area. So, while it's easy to make circles (and you don't even need a computer to make them), the results will be poor, at best.

You could conduct a survey of library patrons to find out where they live. Then you can plot the dots and try to create a choropleth map^{5}. The downside to this is that you'll have to make sure people only respond at one library. Other problems include the fact that some people don't like surveys for whatever reason and will not answer or will lie about where they live. Surveys are also expensive and take time to conduct. You'd also have to consider the time of year because there's probably a seasonality. You might also receive responses from people who go to a library that's on the way to work, rather than the one that's nearest to their home. A well-run survey can give you very high quality data, but at the cost of both time and money.

Or you could create Thiessen polygons for each library. This could reveal libraries where you might want to run a survey to better define the catchment areas for those branches in particular. Actually performing a Thiessen polygon operation in a geographic information system (GIS)^{6} is a relatively trivial matter once you have your points, so you're probably going to maximise the returns on time and money, both of which will be small compared to running a survey. And if you want to improve on the quality of the catchment area depictions, you could impose your city's road network so that the catchment areas are defined by roads rather than pure polygon boundaries.

**Technical Discussion of the Theory of Thiessen Polygons**

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Boots (see Bibliography below) formally describes Thiessen polygons thus:

Consider a set, *S*, of *n* labelled points in the plane, where

S = { p_{1}, p_{2}, … p_{n} }

With each point, p_{i}, in *S* we associate all locations, x, in the plane that are closer to p_{i} than to any other point, p_{j}, in S(j≠i). The result is to create a Thiessen polygon, P_{i}. More formally, if d(x,i) is the Euclidean distance from x to p_{i} then

P_{i} = { x|d(x,i) ≤ d(x,j); j ∈ S, j ≠ i}

It is possible that x is equidistant from a pair of points, in which case it will lie on the boundary of P_{i}. In addition, x may be equidistant from three or more points so that it forms one of the vertices of P_{i}.

If Thiessen polygons are created for all points in *S*, then the resulting set of polygons

{ P_{1}, P_{2}, … P_{n} },

forms a unique, contiguous, space-exhaustive tessellation known as the Thiessen (Voronoi) diagram of S, V(S).

In other words, the area contained in a Thiessen polygon is likely to be more representative of the point on which the polygon is based than of any other point in the set.

**Further Reading**

Should you wish to learn more about Thiessen polygons, here is a short biplanebliography you may find useful:

- BN Boots,
*Quantitative Analysis in Geography*, Waterloo lectures in geography; v.3 (Waterloo: University of Waterloo) - Nicholas R Chrisman
*Exploring Geographic Information Systems* - Michael N DeMers.
*Fundamentals of Geographic Information Systems* - DJ Maguire, Michael F Goodchild and David W Rhind
*Geographical Information Systems: Principles and Applications*