TOPICS
Search

Stereographic Projection


StereographicProjection

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.

StereographicProjections

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.

StereographicProjectionMap

The transformation equations for a sphere of radius R are given by

x=kcosphisin(lambda-lambda_0)
(1)
y=k[cosphi_1sinphi-sinphi_1cosphicos(lambda-lambda_0)],
(2)

where lambda_0 is the central longitude, phi_1 is the central latitude, and

 k=(2R)/(1+sinphi_1sinphi+cosphi_1cosphicos(lambda-lambda_0)).
(3)

The inverse formulas for latitude phi and longitude lambda are then given by

phi=sin^(-1)(coscsinphi_1+(ysinccosphi_1)/rho)
(4)
lambda=lambda_0+tan^(-1)((xsinc)/(rhocosphi_1cosc-ysinphi_1sinc)),
(5)

where

rho=sqrt(x^2+y^2)
(6)
c=2tan^(-1)(rho/(2R))
(7)

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

 R=(R_ecosphi)/((1-e^2sin^2phi)coschi),
(8)

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 Projection

Explore with Wolfram|Alpha

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 Projection

Cite this as:

Weisstein, Eric W. "Stereographic Projection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StereographicProjection.html

Subject classifications