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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

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