Properties
As defined in the preceding section, the Lambert azimuthal projection of the unit sphere is undefined at (0, 0, 1). It sends the rest of the sphere to the open disk of radius 2 centered at the origin (0, 0) in the plane. It sends the point (0, 0, -1) to (0, 0), the equator z = 0 to the circle of radius centered at (0, 0), and the lower hemisphere to the open disk contained in that circle.
The projection is a diffeomorphism (a bijection that is infinitely differentiable in both directions) between the sphere (minus (0, 0, 1)) and the open disk of radius 2. It is an area-preserving (equal-area) map, which can be seen by computing the area element of the sphere when parametrized by the inverse of the projection. In Cartesian coordinates it is
This means that measuring the area of a region on the sphere is tantamount to measuring the area of the corresponding region the disk.
On the other hand, the projection does not preserve angular relationships among curves on the sphere. No mapping between a portion of a sphere and the plane can preserve both angles and areas. (If one did, then it would be a local isometry and would preserve Gaussian curvature; but the sphere and disk have different curvatures, so this is impossible.) This fact, that flat pictures cannot perfectly represent regions of spheres, is the fundamental problem of cartography.
As a consequence, regions on the sphere may be projected to the plane with greatly distorted shapes. This distortion is particularly dramatic far away from the center of the projection (0, 0, -1). In practice the projection is often restricted to the hemisphere centered at that point; the other hemisphere can be mapped separately, using a second projection centered at the antipode.
Read more about this topic: Lambert Azimuthal Equal-area Projection
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