Wikipedia:Reference desk/Archives/Mathematics/2024 September 23

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September 23

Converting map scale into height and width

How can I calculate the height and width in meters of a future geographic map of the world based on its desired scale in cm and km? 212.180.235.46 (talk) 15:35, 23 September 2024 (UTC)

For a map of the world, you will have to specify which map projection it will use, and then to which part(s) of the projection the scale is to apply (and for some projections, how much of the globe to include). Mapping a spherical surface onto a plane surface cannot be done with a constant scale factor. Eg: you might specify a Mercator projection, between latitudes +/-85deg, with a scale of 1:10,000,000 at the equator, which would give you a map about 4m wide along the equator, and (judging by the illustration for Mercator) slightly more than that high. Other projections will vary considerably. -- Verbarson  ^talk_edits 18:07, 23 September 2024 (UTC)

My own map with van der Grinten projection and equatorial scale of 1:20 000 000 for example is 1.18 m high and 1.94 m wide. Out of curiosity, I started to think about dimensions of imaginative humongous world maps. With that as benchmark, if my calculations are correct, a 1:160 000 scale world map, for example, would be 147 m long and 242 m wide. 212.180.235.46 (talk) 20:35, 23 September 2024 (UTC)

The scale of a map is the ratio between the distance between two points on the map and that between the corresponding points on the ground. Any map projection necessarily distorts the distances, so the scale you get if you take New York and San Francisco as the two points is generally substantially different from what you get if you take Berlin and Moscow. For maps of the whole Earth it is non-trivial to decide on a "nominal scale". In spite of the name, the van der Grinten projection is not a geometric projection in which points on the map are found by following rays projected along a straight line out of a shrunk globe. The equatorial and central-meridional scales are well defined, though. The length ℓ_eq of the equator is approximately 40,000 km. If the equatorial scale is chosen to be 1 : S_eq, the width of the map is ℓ_eq / S_eq. So for S_eq = 160,000 this comes out as about 250 m. The length ℓ_mer of a meridian, measured from one pole to the other, is about 40,000 km. Letting S_cm stand for the central-meridional scale factor, the height of the map will be ℓ_mer / S_cm. If these scale factors are chosen to be equal, the map will have a height of half its width. For a circular map as shown in the article, S_cm = 2 S_eq.  --Lambiam 10:14, 24 September 2024 (UTC)