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On Natural Earth they offer maps at various resolutions. What does 1:10,000,000 mean? Does it translate to the number of points per square meter or mile?

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It is the number of meters on the world to the number of meters on the paper. Or any other unit you pick provided that is the same unit on both sides. The wikipedia page explains it pretty well.

So in your 1:10M example it means that 1 cm on the page is 100 km in the real world (if I've done the maths right in my head :-) For a computer data set like Natural Earth the scale is more of a hint about how much detail you can expect.

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What does this mean when talking about vector data? –  Garrett Hall Mar 7 '12 at 16:23
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It means the same thing. Bear in mind that scales on a computer screen are usually an approximation, since you can change the physical size of a screen without changing its resolution (think: projector screens) –  mwalker Mar 7 '12 at 16:36
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@GarrettHall In your example 1:10,000,000 means that 1 unit on your map (screen) is equal to 10,000,000 units on Natural Earth. i.e. 1m on map = 10,000,000m on Natural Earth. –  Alex Markov Mar 7 '12 at 17:08
    
Garrett, that's an excellent question, because vector data have no inherent resolution. In practice, the "resolution" of a vector dataset refers to the resolution with which it was created. The archetypical example consists of placing a paper map (which does clearly have a resolution) on a digitizer and manually plotting points to mark features on it. A steady hand and accurate eye will usually place marks with an accuracy of 0.5 mm. This is the basis for US National Map Accuracy Standards. –  whuber Oct 7 at 19:20

Could your question be reformulated to: Features of which (real world) size are correctly represented in this data set if its resolution is 1:10m? I had that question to the Natural Earth data set.

After talking to colleagues and finally reading this post Appropriate pixel size when converting vector to raster I made up my answer. I am not sure, if it is right. Probably it is wrong because the number I get (see below) is far to large. So, please correct me.

I think the scale refers to the resolution of the material from which the data set was created (satellite pictures (?), existing maps, ... of 1:10m resolution). There are several data sources from which Natural Earth data set was created. The most coarse resolution of all used data sources as 1:10m. As written in the release notes (http://www.naturalearthdata.com/blog/miscellaneous/natural-earth-v2-0-0-release-notes/) coastlines have a finer resolution than 1:10m

We could calculate the accuracy of our data set by this approach:

  • We print the data set in the resolution of 1:10m = 1:10^7 on paper.
  • We can recognise features by an accuracy of 0.3 mm (does not matter whether +/- 0.3 mm or +/- 0.15 mm) ("From a cartographic point of view, it is commonly assumed that the human perception of a line position is around 0.3 mm.", see linked post).
  • 0.3 mm * 10^7 = 3 km
  • Thus, features below about 3 km in extend are not necessarily represented correctly.
  • However, there are object types (e.g. coastlines) in the Natural Earth data set which have a finer resolution than 1:10m. Thus, features in coastlines below a size of 3 km can be correctly represented.
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Your calculations are right. It is interesting how a simple quantitative analysis can inform and even alter how you think about GIS data and maps, isn't it? Note, though, that "accurate" and "correct" may (and usually do) mean different things. "Accurate" means how well you can compute the true position of a point based on its representation on a map. "Correct" may refer to whether that point is shown to be located in its proper country, or on the proper side of a nearby river, or within the right distance of a nearby landmark, for instance. Maps can be highly inaccurate yet correct. –  whuber Oct 7 at 19:24

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