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Stereographic Projection

\begin{figure}\begin{center}\BoxedEPSF{maps/ster.epsf scaled 320}\end{center}\end{figure}

A Map Projection in which Great Circles are Circles and Loxodromes are Logarithmic Spirals.

$\displaystyle x$ $\textstyle =$ $\displaystyle k\cos\phi\sin(\lambda-\lambda_0)$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle k[\cos\phi_1\sin\phi-\sin\phi_1\cos\phi\cos(\lambda-\lambda_0)],$ (2)

k={2\over 1+\sin\phi_1\sin\phi+\cos\phi_1\cos\phi\cos(\lambda-\lambda_0)}.
\end{displaymath} (3)

The inverse Formulas are given by
$\displaystyle \phi$ $\textstyle =$ $\displaystyle \sin^{-1}\left({\cos c\sin\phi_1+{y\sin c\cos\phi_1\over\rho}}\right)$ (4)
$\displaystyle \lambda$ $\textstyle =$ $\displaystyle \lambda_0+\tan^{-1}\left({x\sin c\over\rho\cos\phi_1\cos c-y\sin\phi_1\sin c}\right),$ (5)

$\displaystyle \rho$ $\textstyle =$ $\displaystyle \sqrt{x^2+y^2}$ (6)
$\displaystyle c$ $\textstyle =$ $\displaystyle 2\tan^{-1}({\textstyle{1\over 2}}\rho).$ (7)

See also Gall's Stereographic Projection


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.

© 1996-9 Eric W. Weisstein