Mercator Projection

The following equations place the x-Axis of the projection on the equator and the y-Axis at Longitude $\lambda_0$ , where $\lambda$ is the Longitude and $\phi$ is the Latitude.

$\displaystyle x$	$\textstyle =$	$\displaystyle \lambda-\lambda_0$	(1)
$\displaystyle y$	$\textstyle =$	$\displaystyle \ln[\tan({\textstyle{1\over 4}}\pi+{\textstyle{1\over 2}}\phi)]$	(2)
	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}\ln\left({1+\sin\phi\over 1-\sin\phi}\right)$	(3)
	$\textstyle =$	$\displaystyle \sinh^{-1}(\tan\phi)$	(4)
	$\textstyle =$	$\displaystyle \tanh^{-1}(\sin\phi)$	(5)
	$\textstyle =$	$\displaystyle \ln(\tan\phi+\sec\phi).$	(6)

$\displaystyle \phi$	$\textstyle =$	$\displaystyle 2\tan^{-1}(e^y)=\tan^{-1}(\sinh y)=\mathop{\rm gd}\nolimits y$	(7)
$\displaystyle \lambda$	$\textstyle =$	$\displaystyle x+\lambda_0,$	(8)

An oblique form of the Mercator projection is illustrated above. It has equations

$\displaystyle x$	$\textstyle =$	$\displaystyle {\tan^{-1}[\tan\phi\cos\phi_p+\sin\phi_p\sin(\lambda-\lambda_0)]\over\cos(\lambda-\lambda_0)}$	(9)
$\displaystyle y$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}\ln\left({1+A\over 1-A}\right)= \tanh^{-1} A,$	(10)

$\displaystyle \lambda_p$	$\textstyle =$	$\displaystyle \tan^{-1}\left({\cos\phi_1\sin\phi_2\cos\lambda_1-\sin\phi_1\cos\... ...ver \sin\phi_1\cos\phi_2\sin\lambda_2-\cos\phi_1\sin\phi_2\sin\lambda_1}\right)$
			(11)
$\displaystyle \phi_p$	$\textstyle =$	$\displaystyle \tan^{-1}\left({-{\cos(\lambda_p-\lambda_1)\over\tan\phi_1}}\right)$	(12)
$\displaystyle A$	$\textstyle =$	$\displaystyle \sin\phi_p\sin\phi-\cos\phi_p\cos\phi\sin(\lambda-\lambda_0).$	(13)

$\displaystyle \phi$	$\textstyle =$	$\displaystyle \sin^{-1}\left({\sin\phi_p\tanh y+{\cos\phi_p\sin x\over\cosh y}}\right)$	(14)
$\displaystyle \lambda$	$\textstyle =$	$\displaystyle \lambda_0+\tan^{-1}\left({\sin\phi_p\sin x-\cos\phi_p\sinh y\over \cos x}\right).$	(15)

There is also a transverse form of the Mercator projection, illustrated above. It is given by the equations

$\displaystyle x$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}\ln\left({1+B\over 1-B}\right)=\tanh^{-1} B$	(16)
$\displaystyle y$	$\textstyle =$	$\displaystyle \tan^{-1}\left[{\tan\phi\over \cos(\lambda-\lambda_0)}\right]-\phi_0$	(17)
$\displaystyle \phi$	$\textstyle =$	$\displaystyle \sin^{-1}\left({\sin D\over\cosh x}\right)$	(18)
$\displaystyle \lambda$	$\textstyle =$	$\displaystyle \lambda_0+\tan^{-1}\left({\sinh x\over\cos D}\right),$	(19)

$\displaystyle B$	$\textstyle \equiv$	$\displaystyle \cos\phi\sin(\lambda-\lambda_0)$	(20)
$\displaystyle D$	$\textstyle \equiv$	$\displaystyle y+\phi_0.$	(21)

Finally, the ``universal transverse Mercator projection'' is a Map Projection which maps the Sphere into 60 zones of 6° each, with each zone mapped by a transverse Mercator projection with central Meridian in the center of the zone. The zones extend from 80° S to 84° N (Dana).

Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 38-75, 1987.