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A cartography question: you open an atlas and usually at the beginning of it there are some world maps showing various thematics. Let's say the world map uses Robinson projection and it says it has a scale of 1 : 120,000,000.

How is this scale determined? Are there any standard practices (measuring scale ratio of 10° longitude on equator or something similar)? Or is this an imprecise science left to each cartographer to figure out?

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are you asking why did the cartographer choose 1:120,000,000 instead of 1:100,000,000 or some other scale? – toms Jan 19 '14 at 21:41
No, let me rephrase it: I'm asking how do you determine what the scale of a (say) world map is when there's no scale value printed on the map. – Igor Brejc Jan 19 '14 at 21:45
Maybe you should have a look at the definition of map scales? – Martin Jan 20 '14 at 6:47

In order to compute the scale, you need to work along the standard lines (which are not necessarily straight lines on the map). With cylindrical and conical projections, those standard lines are located at the intersection/tangent with the sphere/ellipsoid. For azimuthal equidistant, those all straight lines that converge to the points of tangency; for the sinusoidal projection, linear scale is true for the equator and the prime meridian; van der grinten, linear scale is true at the equator; for Eckert, it is true at two parallels ...

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Does each projection has allways one (or more) standard lines? So does non-equidistant projections have one (or more) standard lines? – Jens Jan 20 '14 at 8:55
No, not all projection have standard lines (for instance, Azimuthal stereographic or azimuthal gnonomic do not), but it is most often the case that you have at least one standard line. Note that equidistant projection have an infinity of standard lines, but this does not mean that the scale is true in all directions (for instance, conic equidistant has a true linear scale for meridians (infinity of lines) + 1 ( for tangent projection) or 2 (for secant projection) parallels . – radouxju Jan 20 '14 at 10:26

Answering my own question. After thinking about it, the answer seems pretty obvious: map projection formulas. Let's take Natural Earth projection as an example.

In order to project the map to a 1 : 1 scale, you multiply the projected coordinates with Earth's radius (cca. 6371008 m). Such an (imaginary) map has a width of (roughly) 35,000 km (I'm using the test data provided by the author of Natural Earth projection). So in order to show such a map on a piece of paper with a width of 20 cm, you need to scale it down by 180,000,000. And this is the scale of the final map.

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