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I had an idea for a kind of visualisation, and I'd like to know if it's been done before, what it's called etc.

Basically it would visualise a single numerical quantity that varies greatly, at relatively few sample points. For instance, population sizes of cities. It would probably also work well for signal strength from radio transmitters.

To calculate the visualisation using a raster scan, given n(i) is the population of city i:

  1. Assign every city a unique colour. (Or, better, a small number using eg 5-color algorithm.)
  2. At every (X,Y), calculate n(i)/d for every city i where d is the distance from the point to i. (Perhaps something like log(n(i))/d would be better.)
  3. Choose the city that has the highest value, and color the point accordingly.

The end result would be that a city with a big population would have a large circle around it, interrupted when it runs into smaller towns. A very small town might be just a small dot within a much larger circle.

Obviously it would look nicer to calculate this using vectors, but I have no idea how.

Is this a known thing? Any libraries, tools, techniques to produce it?

EDIT

Here's the closest thing I can find, using the tip of searching for "influence map". (Unfortunately that term is also used to describe an unrelated meme in the deviantart online art community...):

enter image description here

I'm imagining something with much simpler, more geometric, borders between points though.

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    A post about a new visualization process without any visual aids is definitely going to rate an "unclear what you're asking" – Vince Nov 25 '14 at 1:46
  • Check out "influence maps". – mkennedy Nov 25 '14 at 18:35
  • @Vince, really? I've explained in quite a lot of detail exactly how the visualisation would work. Are you asking for a hand drawn sketch or something? What extra information would you like? – Steve Bennett Nov 25 '14 at 22:04
  • Yes, a sketch would be valuable, more so than something very complicated at low resolution. – Vince Nov 25 '14 at 22:26
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    Interesting idea! Search also for "proximal polygons", "Thiessen polygons", or as "Dirichlet regions" (different names for same thing). Your situation appears to call for a combination of the above proximal maps, mkennedy's "influence maps", and Reilly's "law of retail gravitation" (or simply the "gravity model"). – Martin F Nov 26 '14 at 6:24
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Here's one ESRI plugin for "weighted voronoi diagrams". @martinf pointed me in the right direction.

An ArcGIS Extension for Generating and Updating Ordinary and Multiplicatively Weighted Voronoi Diagrams for Points, Lines, and Polygons

enter image description here

Unfortunately I don't have access to ArcGIS, so this particular solution isn't much good to me. It's still useful to see that "it is a thing".

Ah, I see this has even been addressed before on gis.se: How can I create weighted Thiessen polygons?

enter image description here

That last one also points to a paper on a vector algorithm (as opposed to the much simpler raster process).

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