The azimuthal equidistant projection

Greeks were the first to describe the Earth as a globe. Their philosophers started by imagining the sun in the center of the universe. They used the globe as a symbol of the whole cosmos. Since then, kings were represented with the globe in their hands to show their power. The globe was visible together with statues and pictures, an example of which is the Atlante Farnese preserved in Naples.

This is a statue out of the second century, representing the celestial sphere as a globe. By the way, the first preserved terrestrial globe map is the Erdapfel, created by Martin Behaim in 1492. The American continent was not represented, but the Earth was, however, portrayed as a globe. The idea of a spherical Earth became more and more widespread. And, during the age of the great sailors and the oceanic travels, having the Earth represented on maps developed into a must.

The Mercator projection map

This meant that the sphere had to be drawn on flat papers. Mercator is well known for having created one of the most famous of these maps. It is known as the Mercator projection map of 1569. It was a “conformal” map studied for navigation. Conformal means that the projection preserves the angles as on the globe and this is a useful characteristic during the navigation.

Conformal maps

A conformal map is characterized by a big deformation of the Earth as the latitude grows. A whatever projection of the Earth on a map induces deformations due to the double curvature of the globe. A cylinder has a single curvature and can be easily projected on a plane, a sphere is different. This is true also for the Mercator projection. A well-known problem of this projection, which is so widespread all over the world, is that it enlarges a lot the northern nations, leaving other zones, like Africa, smaller in size than they are in reality. And, as you can clearly guess, the geographical representation of the world has been used as a sort of propaganda tool for social and political purposes.

A big original flaw

As you can easily understand, all flat maps of the Earth present a big original flaw: the globe. Since all geographers and cartographers considered, while performing their job, the Earth as a globe, they portrayed the Earth over the sphere. All subsequent maps are only projection transferred from the globe over a flat plane, or over a single curvature surface like a cone or a cylinder, that can be easily flattened after being projected from the globe.

Since all maps are obtained this way, with a projection, large deformations are transferred in all of them. The Earth is flat. A big misrepresentation is introduced when the earth is regarded as a globe and a second inaccuracy is introduced when the globe is flattened over a map. Often the second trespass makes the first even bigger and it becomes impossible to go back to the reality.

The azimuthal equidistant projection

Let’s try to comment a little the map that is often used to portray the Earth: the azimuthal equidistant projection centered on the North Pole.

It is clear that, among all projections, this is the more suitable one to show the disposition of the lands. The south pole is all around the Earth, the north pole is in the center, the angles between meridians are preserved and all distances measured starting from the north pole are equal to those on the globe. In comparison with the globe, big deformations of the southern part of the Earth are manifest. The South Pole circumference is enormous in comparison with the globe south pole, but this is correct in many flat-earther’s eyes. But, the one who knows a little the real measures of the Earth recognizes that the distances on the southern hemisphere of the globe are not correct. Surprisingly, this map, notwithstanding all its merits, is off target. It is a simple projection from the globe.

Maps of different models

Maps can be:

-conformal, if directions from one point are preserved; the angle between two directions in the map is the same in the globe;

-equivalent, if the areas in the map are the same in the globe. The shape will be deformed;

-equidistant, if the distances along some direction are preserved.

The azimuthal equidistant map preserves the angle of meridians (azimuth) but also the distances from the North Pole. This map is drawn by the projecting of all points which develop on a plane tangent on the North Pole of the globe.

Analyzing some inaccuracies

We will now consider the inaccuracy of this projection if compared with our flat Earth model.

In our model, the Cancer tropic is at 6660 km from the North Pole. On the globe, the situation is different and thus also on the azimuthal projection. On the globe, we have for each degree of latitude 111.1 km. the latitude of the tropic is 23.4° that means 66.6° from the pole. The tropic is thus at 66.6*111.1=7400 km from the North Pole. The difference is of 740 km with a 10% error that is not a small fault.

The equator on our model is at 11100km from the pole. On the globe is at 90°x111.1= 10000km. The difference is of 1100km with an 11% error. So the band from the cancer tropic to the equator is a bit larger in our model, 4440km than in the globe: 2600km. Thus, the globe, and consequently the azimuthal projection, compress a lot the band between the first tropic and the equator. On the other hand, it expands the area from the pole to the tropic.

When considering our model, the Capricorn tropic is at 13320 km from the Pole. On the globe, we have to calculate (90°+23.4°)x111.1= 12600km. The difference is of 720km with an error of 5.7%. The band between the two tropics is in our model of 6660 km but only 5200 km on the globe.

On our model, the distance between the Capricorn tropic and the South Pole is again 6660km (19980-13320) while on the globe it is again of 7400 km. (They have to preserve the symmetry).

Australia on a flat earth model

You understand that, when we compare the azimuthal map with our model, we discover a lot of differences in the position of the tropics and of the equator.

We have to admit that the countries between the tropics are in reality larger than in the globe, while the others are narrower. For example, Australia, on a flat earth model, is for sure narrower than on the Globe. But does Australia exist? I’m joking.



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