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)

where

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

(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)

where

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

See also Gall's Stereographic Projection

References

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.