# Create weighted Thiessen polygons?

I have a point shapefile and I create Thiessen (Voronoi) polygons programmatically using this scripting syntax:

``````CreateThiessenPolygons_analysis (in_features, out_feature_class, fields_to_copy)
``````

However, each point is related with an area (i.e. the preferred size of each polygon) and I wish the Thiessen polygons to be weighted based on this field.

Is that possible and how?

Is there any relevant code in VBA?

There are many ways to weight distances for constructing Thiessen polygons. The basic idea in constructing them is based on comparing the distance between an arbitrary point x and two fixed points p and q; you need to decide whether x is "closer" to p than to q or not. To this end--at least conceptually--we consider the distances dp = d(x, p) and dq= d(x, q). The weighting usually occurs in two ways: the points can be given positive numeric weights wp and wq and the distances themselves can be transformed.

To make sense, the transformation (which I will write as f) should increase as the distances increase; that is, f(d') > f(d) whenever d' > d >= 0. Examples of such transformations are f(d) = d+1, f(d) = d^2 (Reilly's Law of Retail Gravitation), f(d) = 1 - 1/d (assuming all distances are less than 1), f(d) = log(d), f(d) = exp(d)-1.

We would then say x is "closer" to p than to q exactly when

f(d(x, p)) / wp < f(d(x, q)) / wq.

Notice the division by the weights, rather than multiplication: this means large weights will tend to "pull in" points at larger distances. You will see this in the running example below.

Here's the beautiful thing, and the whole point of this somewhat abstract exposition: although the resulting Thiessen regions can have complex, extremely difficult to calculate boundaries, they are relatively easy to compute using a grid-based representation. Here's the recipe:

1. For each input point p, compute its Euclidean distance grid [d(p)].

2. Use Map Algebra to apply f and the weights, thereby re-expressing each distance grid as

[fp] = f([d(p)]) / wp.

Here is an example using f(d) = 100 + d^(3/2); the scale is 400 by 600. As f(d) increases the value gets darker. Evidently the distance in this example is with respect to the central red point; the other four points get their separate distance calculations (not shown). The areas of the dots are proportional to their weights, which are 2, 10, 3, 4, and 5.

3. Compute the local minimum of all these grids [fp]. Call this [f]. Here is an example. 4. By comparing [f] to each [fp], to each grid cell assign the identifier of the first p for which [f] >= [fp]. (This can be done in one step with a lowest position operation, for instance.) (I doubt there exists an algorithm anywhere that will compute a vector-format solution for this weighting function f.)

Obviously if you have more than a handful of points p you will script this, and if their number runs into thousands you will probably abandon the attempt as being computationally impracticable (although there are ways to expedite the calculation by tiling it).

Another example, showing Thiessen polygons on an ellipsoid, appears at https://gis.stackexchange.com/a/17377/.

• +1 I never realized how much easier this problem becomes by taking a raster approach. – Kirk Kuykendall Nov 28 '11 at 2:31
• Whuber: Very sophisticated process! However, to focus in my applciation;Each point of my input file represents the approximate centoid of a land parcel. I create using that script line noted above a vector Thiessen polygons file. Each polygon is assigned a space i.e. a size based on the thiessen polygons principle of equal distance of boundaries.On the other hand,each land parel has a predefined size which is provided in the area field; and this is the factor that I want to take into account so as polygons will be proportinally to this factor. Any idea please? – Demetris Nov 28 '11 at 14:13
• I do not understand your remarks, Demetris. They sound like you really want an area cartogram rather than a collection of Thiessen polygons. It would help to explain why you are computing these polygons. What problem are they going to solve? How will they be interpreted? – whuber Nov 28 '11 at 15:30
• Whuber:Each of my points input in the Thiessen polygon process represent the approximate centroid of a new land parcel set. Thus, I create thiessen polygons based on these points representing the shape of of land parcels (one point-one land parcel). I may produce many sets of random land parcels shapes by moving these points to feed my genetic algorithm. The problem is that these generated parcels shapes (i.e.Thiessen polygons) should have a predefined area and I wonder if it is possible to take this into account when utilising the Thiessen polygons operation. I hope that this make sense. – Demetris Nov 28 '11 at 19:10
• What is your genetic algorithm attempting to do? It still sounds like you don't need weighted Thiessen polygons: I believe there exists no possible weighting that will assure the polygons attain prespecified areas or even prespecified relative areas. – whuber Nov 28 '11 at 19:19

What you want is a weighted Voronoi diagram: http://en.wikipedia.org/wiki/Weighted_Voronoi_diagram also know as a circular Dirichlet tessellation when done with multiplicative weights in a 2d plane. Someone seems to have built an arcgis 9 extension to build these: http://arcscripts.esri.com/details.asp?dbid=15481 With a user guide available here http://geography.unt.edu/~pdong/software.htm and a paper published at Dong, P., 2008. Generating and updating multiplicatively weighted Voronoi diagrams for point, line and polygon features in GIS. Computers & Geosciences, Volume 34, Issue 4, Pages 411-421.

There is a recent article on a vector based algorithm (I assume P Dong's algorithm is raster based) for this. http://www.sciencedirect.com/science/article/pii/S0098300411003037 Abstract says c# code is included.

• Blord-castillo: Many thanks for all this information. It is very usefull and I will accept this as a comprehensive answer. However,my new problem is that I wish to run that tool within my code for a number of times by providing inputs like e.g.as the above script line. Is that possible? – Demetris Nov 27 '11 at 17:01