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:
And
For the azimuthal equidistant, an important projection for navigational applications, the distance from the center of the map to any other point is directly proportional to its true radial distance from the center of projection. In the north polar aspect:
The austral aspect is just as easy, with
In the only projection both azimuthal and equal-area, created by Lambert and suitable for world maps, the relative distance of points from the center of the map is progressively reduced in order to keep areal equivalence. Formulas follow from basic integral calculus; first we define an area element on both Earth and map.
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.
The combined map shows both projections virtually identical at latitudes north of 60°N (as predicted by the graph: = 0.534). Beyond that, parallels get closer and closer together in Lambert's half, while remaining equally spaced in the azimuthal equidistant portion. The resulting areal difference is clearly visible in Antarctica.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.
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 |
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
In the equations for Lambert's equal-area azimuthal projection, substitute for :
Now using two trigonometric identities,
the common factor can be expanded:
Finally,
Again, both projections are very similar near the center of projection: the northern and southern portions of Africa join almost seamlessly.