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Gnomonic Projection


GnomonicProjectionFigure

The gnomonic projection is a nonconformal map projection obtained by projecting points P_1 (or P_2) on the surface of sphere from a sphere's center O to point P in a plane that is tangent to a point S (Coxeter 1969, p. 93). In the above figure, S is the south pole, but can in general be any point on the sphere. Since this projection obviously sends antipodal points P_1 and P_2 to the same point P in the plane, it can only be used to project one hemisphere at a time. In a gnomonic projection, great circles are mapped to straight lines. The gnomonic projection represents the image formed by a spherical lens, and is sometimes known as the rectilinear projection.

GnomonicProjection

In the projection above, the point S is taken to have latitude and longitude (lambda,phi)=(0,0) and hence lies on the equator. The transformation equations for the plane tangent at the point S having latitude phi and longitude lambda for a projection with central longitude lambda_0 and central latitude phi_1 are given by

x=(cosphisin(lambda-lambda_0))/(cosc)
(1)
y=(cosphi_1sinphi-sinphi_1cosphicos(lambda-lambda_0))/(cosc),
(2)

and c is the angular distance of the point (x,y) from the center of the projection, given by

 cosc=sinphi_1sinphi+cosphi_1cosphicos(lambda-lambda_0).
(3)

The inverse transformation equations are

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=tan^(-1)rho
(7)

and the two-argument form of the inverse tangent function is best used for this computation.


See also

Stereographic Projection

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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. 164-168, 1987.

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Gnomonic Projection

Cite this as:

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

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