Lambert azimuthal equal-area projection

The Lambert azimuthal equal-area projection is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772. "Zenithal" being synonymous with "azimuthal", the projection is also known as the Lambert zenithal equal-area projection. The Lambert azimuthal projection is used as a map projection in cartography. For example, the National Atlas of the US uses a Lambert azimuthal equal-area projection to display information in the online Map Maker application, and the European Environment Agency recommends its usage for European mapping for statistical analysis and display. It is also used in scientific disciplines such as geology for plotting the orientations of lines in three-dimensional space. This plotting is aided by a special kind of graph paper called a Schmidt net. 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 R3 such that x2 + y2 + z2 = 1.