stereographic projection

Noun

 * 1)  A function that maps a sphere onto a plane; especially, the map generated by projecting each point of the sphere from the sphere's (designated) north pole to a point on the plane tangent to the south pole.
 * 2) * 1974 [Prentice-Hall], Richard A. Silverman, Complex Analysis with Applications, Dover, 1984, Unaltered republication, page 19,
 * Thus we see that stereographic projection establishes a one-to-one correspondence between the extended complex plane and the Riemann sphere.
 * 1) * 1991, E. J. F. Primrose (translator), Simeon Ivanov (editor of translation),, A. A. Tuzhilin, Elements of the Geometry and Topology of Minimal Surfaces in Three-Dimensional Space, , page 67,
 * Proposition 2. The stereographic projection $$\pi_N: S^2 \to Oxy$$ of the sphere $$S^2$$ onto the $$xy$$-plane from the North Pole $$N$$ preserves the angles between tangent vectors. Thus, the coordinates specified on $$S^2\setminus N$$ by the stereographic projection $$\pi_N$$ are isothermal.

Usage notes

 * The plane may be augmented with a point at infinity, onto which the north pole is considered to be projected.
 * The projective plane is standardly (after ) the one tangent to the south pole, but the one through the equator is also often chosen.
 * Circles on the sphere that do not pass through the north pole are projected onto circles on the plane. Circles on the sphere that do pass through the north pole are projected onto straight lines (which can, however, be regarded as generalized circles on the augmented plane).
 * The mapping is conformal (angle-preserving, at the point where curves intersect), but does not preserve distances or areas.