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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)

\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)


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.

© 1996-9 Eric W. Weisstein