Azimuthal equidistant projection

Mathematical definition

A point on the globe is chosen as "the center" in the sense that mapped distances and azimuth directions from that point to any other point will be correct. That point, (φ₀, λ₀), will project to the center of a circular projection, with φ referring to latitude and λ referring to longitude. All points along a given azimuth will project along a straight line from the center, and the angle θ that the line subtends from the vertical is the azimuth angle. The distance from the center point to another projected point ρ is the arc length along a great circle between them on the globe. By this description, then, the point on the plane specified by (θ, ρ) will be projected to Cartesian coordinates: $${\displaystyle x=\rho \sin \theta ,y=-\rho \cos \theta }$$

The relationship between the coordinates (θ, ρ) of the point on the globe, and its latitude and longitude coordinates (φ, λ) is given by the equations: $${\displaystyle \begin{array}{cc}\cos \frac{\rho }{R}& =\sin {\phi }_0\sin \phi +\cos {\phi }_0\cos \phi \cos \left(\lambda -{\lambda }_0\right)\\ \tan \theta & =\frac{\cos \phi \sin \left(\lambda -{\lambda }_0\right)}{\cos {\phi }_0\sin \phi -\sin {\phi }_0\cos \phi \cos \left(\lambda -{\lambda }_0\right)}\end{array}}$$

When the center point is the north pole, φ₀ equals ${\displaystyle \pi /2}$ and λ₀ is arbitrary, so it is most convenient to assign it the value of 0. This assignment significantly simplifies the equations for ρ_u and θ to: $${\displaystyle \rho =R\left(\frac{\pi }{2}-\phi \right),\theta =\lambda }$$

Limitation

With the circumference of the Earth being approximately 40,000 km (24,855 mi), the maximum distance that can be displayed on an azimuthal equidistant projection map is approximately 20,000 km (12,427 mi), half the circumference. For distances less than about 10,000 km (6,214 mi) distortions are minimal visually. For distances about 10,000–15,000 km (6,214–9,321 mi) the distortions are visually moderate. Distortions become severe when the distances are greater than about 15,000 km (9,321 mi).

If the azimuthal equidistant projection map is centered about a point whose antipodal point lies on land and the map is extended to the maximum distance of 20,000 km (12,427 mi) the antipode point smears into a large circle. This is shown in the example of two maps centered about Los Angeles, and Taipei. The antipode for Los Angeles is in the south Indian Ocean hence there is not much significant distortion of land masses for the Los Angeles centered map except for East Africa and Madagascar. On the other hand, Taipei's antipode is near the Argentina–Paraguay border, causing the Taipei centered map to severely distort South America.

Applications

Azimuthal equidistant projection maps can be useful in terrestrial point to point communication. This type of projection allows the operator to easily determine in which direction to point their directional antenna. The operator simply finds on the map the location of the target transmitter or receiver (i.e. the other antenna being communicated with) and uses the map to determine the azimuth angle needed to point the operator's antenna. The operator would use an electric rotator to point the antenna. The map can also be used in one way communication. For example if the operator is looking to receive signals from a distant radio station, this type of projection could help identify the direction of the distant radio station. In order for the map to be useful, the map should be centered as close as possible about the location of the operator's antenna.

Azimuthal equidistant projection maps can also be useful to show ranges of ballistic missiles, as demonstrated by the map below. The map is centered on North Korea, along with concentric circle overlays showing the different ranges of the country's missiles.