Gnomonic Projection

$\begin{figure}\begin{center}\BoxedEPSF{maps/gnom.epsf scaled 400}\end{center}\end{figure}$

A nonconformal projection from a Sphere's center in which Orthodromes are straight Lines.

$\displaystyle x$	$\textstyle =$	$\displaystyle {\cos\phi\sin(\lambda-\lambda_0)\over\cos c}$	(1)
$\displaystyle y$	$\textstyle =$	$\displaystyle {\cos\phi_1\sin\phi-\sin\phi_1\cos\phi\cos(\lambda-\lambda_0)\over\cos c},$	(2)

where

$\begin{displaymath} \cos c=\sin\phi_1\sin\phi+\cos\phi_1\cos\phi\cos(\lambda-\lambda_0). \end{displaymath}$

(3)

The inverse Formulas are

$\displaystyle \phi$	$\textstyle =$	$\displaystyle \sin^{-1}(\cos c\sin\phi_1+y\sin c\cos c\cos\phi_1)$	(4)
$\displaystyle \lambda$	$\textstyle =$	$\displaystyle \lambda_0+\tan^{-1}\left({x\over \cos\phi_1-y\sin\phi_1}\right).$	(5)

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. 164-168, 1987.