# Azimuthal equidistant projection

The **azimuthal equidistant projection** is an azimuthal map projection. It has the useful properties that all points on the map are at proportionately correct distances from the center point, and that all points on the map are at the correct azimuth (direction) from the center point. A useful application for this type of projection is a polar projection which shows all meridians (lines of longitude) as straight, with distances from the pole represented correctly. The flag of the United Nations contains an example of a polar azimuthal equidistant projection.

## History

While it may have been used by ancient Egyptians for star maps in some holy books,^{[1]} the earliest text describing the azimuthal equidistant projection is an 11th-century work by al-Biruni.^{[2]}

The projection appears in many Renaissance maps, and Gerardus Mercator used it for an inset of the north polar regions in sheet 13 and legend 6 of his well-known 1569 map. In France and Russia this projection is named "Postel projection" after Guillaume Postel, who used it for a map in 1581.^{[3]} Many modern star chart planispheres use the polar azimuthal equidistant projection.

## Mathematical definition

A point on the globe is chosen to be special in the sense that mapped distances and azimuths from that point to any other point will be correct. That point, (*φ*_{1}, *λ*_{0}), 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 is given as *ρ*. By this description, then, the point on the plane specified by (*θ*,*ρ*) will be projected to Cartesian coordinates:

The relationship between the coordinates (*θ*,*ρ*) of the point on the globe, and its latitude and longitude coordinates (*φ*, *λ*) is found as follows.
The great circle distance *ρ* between two points (*φ*_{1}, *λ*_{0}) and (*φ*, *λ*) on the sphere is given by:
^{[4]}

The azimuth from the first to the second point is given by:

When the center point is the north pole, these formulas greatly simplify to:

## Applications

In the case of radio, this projection allows for directional antenna aiming, especially in the case of HF communications. An operator can point the antenna, usually by an electric rotator, simply locating the target in the map and rotating the antenna to the angle indicated by the map. The map should be centered as nearly as possible to the actual antenna location.

## See also

- List of map projections

## References

- ↑ SNYDER, John P. (1997).
*Flattening the earth: two thousand years of map projections*. University of Chicago Press. ISBN 0-226-76747-7. , p.29 - ↑ David A. KING (1996), "Astronomy and Islamic society: Qibla, gnomics and timekeeping", in Roshdi Rashed, ed.,
*Encyclopedia of the History of Arabic Science*, Vol. 1, p. 128–184 [153]. Routledge, London and New York. - ↑ Snyder 1997, p. 29
- ↑ Snyder, John P.; Voxland, Philip M. (1989).
*An Album of Map Projections*. Professional Paper 1453. Denver: USGS. p. 228. ISBN 978-0160033681. Retrieved 2018-03-29.

## External links

- Table of examples and properties of all common projections, from radicalcartography.net
- Online Azimuthal Equidistant Map Generator
- An interactive Java Applet to study the metric deformations of the Azimuthal Equidistant Projection.
- GeographicLib provides a class for performing azimuthal equidistant projections centered at any point on the ellipsoid.
- Animated US National Weather Service Wind Data for Azimuthal equidistant projection.