stereographic projection

English

3d illustration of a stereographic projection

Noun

stereographic projection (plural stereographic projections)

1. (projective geometry, complex analysis, cartography) 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.
   • 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.

   • 1991, E. J. F. Primrose (translator), Simeon Ivanov (editor of translation), A. T. Fomenko, A. A. Tuzhilin, Elements of the Geometry and Topology of Minimal Surfaces in Three-Dimensional Space, American Mathematical Society, 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.

   • 2004, Richard J. Lisle, Peter R. Leyshon, Stereographic Projection Techniques for Geologists and Civil Engineers, 2nd edition, Cambridge University Press, page 40:

     Before we can recognize such patterns confidently, we need to know whether the stereographic projection faithfully represents the true clustering of directions in space.

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 Coxeter) 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.

See also

• inversive geometry
• Riemann sphere

Further reading

• List of map projections on Wikipedia.Wikipedia
• Inversive geometry on Wikipedia.Wikipedia
• Stereographic Projection on Wolfram MathWorld
• Stereographic projection on Encyclopedia of Mathematics