Stereographic projection

A map from a sphere minus one projection point to a plane, formed by extending lines from that point through the sphere onto a tangent plane. It preserves angles and maps circles to circles or lines.

Conformal map: A function that preserves angles between intersecting curves wherever its derivative is nonzero. Stereographic projection is a concrete conformal map between the punctured sphere and the plane.
Complex plane: A plane whose points represent complex numbers, with the real and imaginary parts as Cartesian coordinates. Stereographic projection supplies planar coordinates for points on the Riemann sphere.
Sphere: The three-dimensional surface whose points lie at a fixed distance from a center. The projection begins with a sphere and sends its surface points to a plane.
Mercator projection: A cylindrical map projection that preserves local angles and represents constant compass bearings as straight lines. Both projections are conformal, but Mercator maps the sphere to a cylinder and distorts polar regions.
Riemann sphere: The complex plane together with one point at infinity, identified with a sphere by stereographic projection. This identification turns planar complex numbers into points on the sphere.
Spherical astronomy: The study of celestial positions and motions represented on the celestial sphere. Stereographic charts can represent directions on the celestial sphere in a plane.
Tangent plane: A plane that touches a smooth surface at a point and shares its tangent directions there. The target plane touches the sphere at the point opposite the projection point.
Gnomonic projection: A map projection from the sphere's center onto a tangent plane, with great circles appearing as straight lines. It shares a planar target but uses the sphere's center, not a surface point, as its projection center.
Möbius transformation: A complex function of the form (az+b)/(cz+d), with ad−bc nonzero, extended to the Riemann sphere. Stereographic coordinates make rotations and other sphere symmetries expressible as Möbius transformations.
Planispheric astrolabe: An astronomical instrument that represents the rotating celestial sphere with a rotating planar star map. Its star map uses stereographic projection to flatten the celestial sphere.

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