Definition
To define the Lambert azimuthal projection, imagine a plane set tangent to the sphere at some point S on the sphere. Let P be any point on the sphere other than the antipode of S. Let d be the distance between S and P in three-dimensional space (not the distance along the sphere surface). Then the projection sends P to a point P′ on the plane that is a distance d from S.
To make this more precise, there is a unique circle centered at S, passing through P, and perpendicular to the plane. It intersects the plane in two points; let P ' be the one that is closer to P. This is the projected point. See the figure. The antipode of S is excluded from the projection because the required circle is not unique. The case of S is degenerate; S is projected to itself, along a circle of radius 0.
Explicit formulas are required for carrying out the projection on a computer. Consider the projection centered at S = (0, 0, -1) on the unit sphere, which is the set of points (x, y, z) in three-dimensional space such that x2 + y2 + z2 = 1. In Cartesian coordinates on the sphere and on the plane, the projection and its inverse are then described by
In spherical coordinates on the sphere (with the zenith and the azimuth) and polar coordinates on the disk, the map and its inverse are given by
In cylindrical coordinates on the sphere and polar coordinates on the plane, the map and its inverse are given by
The projection can be centered at other points, and defined on spheres of radius other than 1, using similar formulas.
Read more about this topic: Lambert Azimuthal Equal-area Projection
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