Stereographic Projection
A map projection obtained by projecting points P on the surface of sphere from the sphere's north pole N to point P^' in a plane tangent to the south pole S (Coxeter 1969, p. 93). In such a projection, great circles are mapped to circles, and loxodromes become logarithmic spirals.
Stereographic projections have a very simple algebraic form that results immediately from similarity of triangles. In the above figures, let the stereographic sphere have radius r, and the z-axis positioned as shown. Then a variety of different transformation formulas are possible depending on the relative positions of the projection plane and z-axis.
The transformation equations for a sphere of radius R are given by
where lambda_0 is the central longitude, phi_1 is the central latitude, and
The inverse formulas for latitude phi and longitude lambda are then given by
where
and the two-argument form of the inverse tangent function is best used for this computation.
For an oblate spheroid, R can be interpreted as the "local radius," defined by
where R_e is the equatorial radius and chi is the conformal latitude.
See also
Gnomonic Projection, Lambert Azimuthal Equal-Area Projection, Map Projection, Vertical Perspective ProjectionExplore with Wolfram|Alpha
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References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 93 and 289-290, 1969.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 150-153, 1967.Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 154-163, 1987.Referenced on Wolfram|Alpha
Stereographic ProjectionCite this as:
Weisstein, Eric W. "Stereographic Projection." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StereographicProjection.html