The map at the beginning of this thread is a Polar Azimuthal Equidistant projection map. Its principal features are that, FROM the center point (here, the North Pole) the direction and distance to any other place is correctly depicted. This projection and the mathematics behind it has been in use for centuries. This one here is centered on the North Pole, but an Azimuthal Equidistant map can be centered on any spot on Earth to show, e.g., how antennae should be turned to maximize radio signals to/from some other location. (Here is an Azimuthal Equidistant map centered on Delhi, India:
[url=http://www.indiandefencereview.com/spotlights/poor-mapping-by-military-survey/]http://www.indiandefencereview.com/spotlights/poor-mapping-by-military-survey/ [/url].) This projection is the one used in the UN emblem - originally it was turned so the United States appeared right-side-up as would be extremely familiar to Americans, but after a few years the UN turned the map so that the more neutral spot of Zero longitude is at the middle. There is more than one Azimuthal Equidistant projection, with very similar but not identical results.
But the Azimuthal Equidistant map is not reliable for shapes or area, especially as the distance from the center point increases, nor are the distances or directions between any two non-center points reliable (esp as distance increases between the two non-center points). Here in the Polar version, the East-West dimension of Australia seems to be three times the North-South dimension - and Australia, east-west, appears to be much larger than the east-west dimension of the US. In reality the the ratio of Australia is more like 5:4 and Australia and the US are about equally wide, east-west.
On this Polar map, the trip along the entire coastline of Antarctica appears to be prohibitively long, the longest journey possible on Earth. Yet, ships and planes (and even birds) have traveled along the entire coastline and the clock and fuel gauge show it was not as arduous as the map makes it appear. The coastline's 11,165 miles is less than the coastline of South America (19,325 miles) which ships from the east/west of the US had to travel before the building of the Panama Canal. On the map, the distance between Buenos Aires and Perth appears to be prohibitive but the actual distance (7836 miles) is just slightly more than a NYC/Paris round-trip (7274 miles).
This is an Azimuthal Equidistant map with Mecca as its center, showing the direction and distance FROM Mecca to any non-center point.
https://anyasword.files.wordpress.com/2011/01/mecca-centred-map.jpg Having mentioned the Azimuthal Equidistant map and its characteristics, there is a rather bizarre projection, the Retro-Azimuthal Equidistant, in which the
direction FROM any non-center point TO the center point is correct (the accuracy of the distance is somewhat less crucial). This is sometimes called the Mecca map because very often centered on Mecca to show the reader in which direction he must face when praying wherever (away from Mecca) he may be.
https://en.wikipedia.org/wiki/Craig_retroazimuthal_projection (unfortunately, this had to replace a more interesting interactive map that vanished from the internet). (In an interactive Hammer Retroazimuthal Equidistant projection, the distortion becomes extreme as the distance from Mecca increases:
[url=https://www.jasondavies.com/maps/hammer-retroazimuthal/]https://www.jasondavies.com/maps/hammer-retroazimuthal/ .)
The Polar Azimuthal Equidistant map seems to be popular among FE enthusiasts, in large part because the Antarctic coastline has the appearance of a pie crust and suggests an explanation why ships don't fall off the edge. But measurements made at Antarctica are very incompatible with the impression made by the map. Here, for example, is an Azimuthal Equidistant map centered on the
South Pole, instead of the more familiar North Pole, ....
http://www.emapsworld.com/images/world-south-pole-azimuthal-equidistant-projection-map.gif ... and, as you can see, Antarctica is much less impressive in size than on most other flat maps where it appears on the outer edges.
Poll