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Azimuthal projections: World Peace, Flat-Earthers and Missile Range Maps

Updated: Jan 21, 2022

OK, we have to get a little bit technical. An azimuthal projection is a map projection where the spherical globe grid is projected onto a flat plane. There are several types of azimuthal projections, but they all have some properties in common: Meridians are shown as straight lines, and parallels as concentric circles. Therefore, lines of latitude and longitude are always crossing at 90 degrees. Azimuthal projections of a hemisphere always form a circular map.

There are 4 important azimuthal projections. Here is how they are defined in the 1982 Geological Survey Bulletin "Map Projections Used by the U.S. Geological Survey", by John P. Snyder:

  1. Orthographic. A true perspective, in which the Earth is projected from an infinite distance onto a plane. The map looks like a globe, thus stressing the roundness of the Earth.

  2. Stereographic. A true perspective in the spherical form, with the point of perspective on the surface of the sphere at a point exactly opposite the point of tangency for the plane, or opposite the center of the projection, even if the plane is secant. This projection is conformal for sphere or ellipsoid, but the ellipsoidal form is not truly perspective.

  3. Lambert Azimuthal Equal-Area. Not a true perspective. Areas are correct, and the overall scale variation is less than that found on the major perspective azimuthals.

  4. Azimuthal Equidistant. Not a true perspective. Distances from the center of the projection to any other point are shown correctly. Overall scale variation is moderate compared to the perspective azimuthals.

Orthographic and Stereographic Azimuthal Projections - Adapted from the USGS Bulletin

The stereographic projection is primarily used for maps of the Polar Regions. The ortographic projection has some broader used. The distortion at the edges of the map is quite severe, but because its form resembles a true '3-D' globe, we almost automatically perceive perspective, especially when an oblique view is used (orthographic globe C at the above image). It is an old projection, already used by Ancient Greeks, in the 2nd century b.c. It regained popularity during WW2, when it was used to demonstrate the global character of the war (more later). The first photographs of the Earth taken from space, like the famous Apollo 17 image "The blue Marble" renewed interest in the orthographic projection, because it shows on a map just how Earth is looking from a remote viewpoint. Nowadays, such a projection is also used in zoomed out views in Google Earth.

1972 NASA "blue Marble" photograph. This is the original photo. It is usually printed in a rotated view, with North up.

The ortographic and stereographic projections are true perspective projections: they can be mechanically constructed. The equidistant and equal-area forms are synthetic constructions. Therefore, they can be used to show the complete world, instead of one hemisphere (although at the cost of much distortion at the edges). However, because of this distortion, often these maps are limited to show only one hemisphere.

Equal-area and equidistant Azimuthal Projections - Adapted from the USGS Bulletin

The Lambert Azimuthal Equal-Area projection adjusts projected distances in order to preserve area. Its equatorial aspect was often used to show the world in two hemispherical maps of the Eastern and Western hemisphere (sphere B of the Lambert equal-area projections showed in the above image).

The equidistant projection's major feature is a concept not too difficult to grasp: distances and directions from one point on the Earth's surface are correctly shown. It was THE projection of the Air-Age (see my previous blog on this cartographic era), for several reasons. First, it showed that the USA were not that isolated from other continents as conventional maps somehow suggested. Secondly, the frequently used North Polar projection allowed accurate depiction of flight routes over the Polar Region.

One of the first ones to use the Azimuthal equidistant projection to communicate these novel insight in political geography was Richard Edes Harrison with his famous 1941 "The World Divided" map made for Fortune Magazine. Very often, these kinds of projections were chosen to communicate more than plain geography. They were often intended to influence public opinion, and view-points were chosen to underline a certain point. They often are beautiful examples of what is called persuasive cartography.

Richard Edes Harrison - The World Divided - from David Rumsey Map Collection

Another example is the 1943 Hammond Air-Age Map of the World shown below. Notice that in the upper corners the Western and Eastern hemisphere are shown in the Lambert Azimuthal Equal-Area projection. In this map, the complete world is shown, resulting in Antarctica shown as a ring of ice surrounding the Oceans.

Hammond's Air-Age Map of the World

A flat Earth surrounded by a wall of ice. Sounds familiar, doesn't it. It is the believe of members of the Flat-Earth Society. This pseudo-scientific movement thrives on conspiracy theories and anti-government sentiments. One of the first to advocate the theory of a flat earth was 19th Centruy writer and inventor Samual Robotham, who published his believes in his book Zetetic Astronomy: Earth Not a Globe in 1865.

Diagram of Earths's surface - From Samual Robotham: Zetetic Astronomy: Earth Not a Globe, 1865

And there is of course another very familiar image in our collective minds: the symbol of the United Nations. The story behind this symbol deserves a blog post on its own, which will follow soon.

United Nations flag with Earth emblem

Most azimuthal equidistant maps we see, are centered on the North Pole. We do however have to realize that the only point from were distances are reliably shown is from the center. Of course, any place on Earth can be chosen as the center of the projection. And luckily, there is a wonderful website where you can do this yourself. Reasons of making an azimuthal equidistant map are to show what places can be reached by plane from a certain point, or which way to direct your antenna, if you are an radio-amateur.

There were however more sinister uses of this projection: one can project target sites, or bomber and missile ranges in easy to grasp and powerful images. An equidistant projection has the advantage of ranges being projected as circles.

In the 1943 Target Tokyo map, made for the US Army by F.E. Manning (image from David Rumsey Map Collection) , the image is centered on Tokyo. The distance from Tokyo is given in 500 mile-radius. But if one would like to measure the distance between any other given place of this map and Tokyo, one could use the 'tape-measure' provided at the bottom of the map.

Target Tokyo Map - From David Rumsey Map Collection

But it is not only used to show how 'close' we are to the enemy, it can also be used to show how 'close' the enemy is to us. The map below dates from the Cuban Missile Crisis. It shows concentric circles around San Diego de los Benos. The range of the missiles was approximately 2200 nautical miles, and therefore posed a threat to almost the entire United States.

On the extraordinary website MissileMap, from Alex Wellerstein, we can project missile ranges from several countries. Different types of missiles can be chosen. As almost every dynamic map on the internet, it uses web Mercator. As can be appreciated, the trajectory shows a slight curve, and the range is no longer a perfect circle, but slightly stretched in the North-South position.

Cuban Missile Crisis simulation, from MissileMap

However, with increasing ranges of modern missiles, the images change completely: Let's select North Korea, and choose a long-range missile, in this case a 13.000 kilometer range Hwasong-15.

Hwasong-15 simulation, from MissileMap

Now, the trajectory is evidently curved, and the range is sinusoidal. Somewhat difficult to grasp, if one is not aware of the properties of a Mercator projection. Now, unfortunately, a North-Korea-centered equidistant projection would not be of much help, as can be seen in the image below. The range is a perfect circle, but the range is so immense, that a projection of the complete globe is needed, which results in a huge distortion at the antipode of North-Korea: South America. It is stretched beyond recognition, from 330 degrees tot 155 degrees, at the border of the map.

North Korean missile range, equidistant - image made with website

And this is where we meet a projection we have seen before: the ortographic azimuthal projection. It results in a hemispheral view resembling a real globe (or Google Earth, for the younger generation). Of course, there is distortion, but that does not boggle the mind, because it is perceived as depth.

North Korea missile range map, from

These 3D-like ortographic azimuthal projections are more and more used in 'cartographic journalism' / infographics, because they are easy to read, without the need of much knowledge on cartographic projections. And they appeal to the eye. One should therefore always be aware of the message the cartographer wants to communicate, because sometimes such maps have more to do with propaganda, than with objective information. This projection was popularized during WW2, with Richard Edes Harrison as one of its main advocates. Harrison was a graphic designer by education, and only entered the cartography-business by chance. Especially his WW2 maps communicate strong messages, and can also be seen as persuasive maps, instead of a pure objective form of cartography.

Azimuthal projections are centuries old, and have almost exclusively been used for maps of the Polar region, until they became popular during WW2. This was mainly because of their ability to show the world in an easy to grasp image. It does however confront us again with the fact that every flat projection of the Earth results in distortions. And ironically, the orthographic projection that is most appealing to the eye and mind, is a projection that projects the globe as if it were a globe...

So maybe the message to everyone interested in missile ranges, world peace and the North Pole should be: Buy a globe!

Franklin D. Roosevelt being presented a globe by the U.S. Army, December 1942. FDR Presidential Library

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