|The general case of azimuthal projections is mainly defined by a function that transforms distances from the center of projection T.|
The mathematical development of the azimuthal orthographic projection is purely geometric. Even though several azimuthal projections — like the two explained here — do not follow such a perspective process, all can be reduced to a general pattern.
On a sphere, two angles determine the distance and position of any point relative to the center of projection : for north polar aspects, equals the longitude , while equals the colatitude , or .
On the map, due to the azimuthal property,
The fundamental characteristic of an azimuthal projection is a function which transforms distances from the center of the map, and therefore determines the spacing of parallels in the polar aspects:
|The defining distance function for some azimuthal projections|
The austral aspect is just as easy, with
Let a thin spherical zone be the element, given the colatitude :
The corresponding element on a polar azimuthal map is a ring with area:
For any given colatitude , we want such that the spherical cap bounded by and the disc bounded by on the map have identical areas. This is enough for areal preservation because the scales along circumferences on both the zone and ring, although different, are constant — again, due to the azimuthal property.
A similar sign change applies if the south polar aspect is intended.
|Combined north polar azimuthal equidistant (top) and equal-area (bottom) maps.|
Calculating other aspects for azimuthal maps is possible by introducing coordinate transformations and rotations in space. However, the important equatorial aspect can be obtained in a more direct way, using two properties of triangles on a spherical surface.
|The laws of sines and cosines for spherical triangles|
|Given the angles , , on a spherical triangle's vertices, and the corresponding angles , , between edges connecting triangle vertices and the center of sphere :|
|Law of sines|
|Law of cosines|
|A triangle on the sphere is defined by the center of projection T, the point to be projected P and an equatorial point with the same longitude.|
On the equatorial aspect, the center of projection lies on the intersection of the Equator and an arbitrary central meridian. It is one vertex of a spherical triangle; the second one is projected point , and the third one lies on the Equator at the same longitude as . The respective angles at the center of the sphere are , and , and we want , the angle at which corresponds to in the first diagram.
If , but
Now using two trigonometric identities,
the common factor can be expanded:
|Combined azimuthal equidistant (top) and equal-area (bottom) equatorial maps.|