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 proportionally 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. While it may have been used by ancient Egyptians for star maps in some holy books, the earliest text describing the azimuthal equidistant projection is an 11th-century work by al-Biruni. An example of this system is the world map by ‛Ali b. Ahmad al-Sharafi of Sfax in 1571. 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. Many modern star chart planispheres use the polar azimuthal equidistant projection. 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, (φ_0, 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 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: 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: When the center point is the north pole, φ0 equals and λ_0 is arbitrary, so it is most convenient to assign it the value of 0.