# 麥卡托投影法

（重定向自墨卡托投影

## 数学计算

{\displaystyle {\begin{aligned}x&=\lambda -\lambda _{0}\\y&=\ln \left(\tan \left({\frac {\pi }{4}}+{\frac {\varphi }{2}}\right)\right)\\&={\frac {1}{2}}\ln \left({\frac {1+\sin(\varphi )}{1-\sin(\varphi )}}\right)\\&=\sinh ^{-1}\left(\tan(\varphi )\right)\\&=\tanh ^{-1}\left(\sin(\varphi )\right)\\&=\ln \left(\tan(\varphi )+\sec(\varphi )\right).\end{aligned}}}

{\displaystyle {\begin{aligned}\varphi &=2\tan ^{-1}(e^{y})-{\frac {\pi }{2}}\\&=\tan ^{-1}(\sinh(y))\\\lambda &=x+\lambda _{0}.\end{aligned}}}

## 公式推导

${\displaystyle {\frac {\partial x}{\partial \lambda }}=\cos(\varphi ){\frac {\partial y}{\partial \varphi }}}$
${\displaystyle {\frac {\partial y}{\partial \lambda }}=-\cos(\varphi ){\frac {\partial x}{\partial \varphi }}}$

x = λ 可知

${\displaystyle {\frac {\partial x}{\partial \lambda }}=1}$
${\displaystyle {\frac {\partial x}{\partial \varphi }}=0}$

${\displaystyle 1=\cos(\varphi ){\frac {\partial y}{\partial \varphi }}}$
${\displaystyle 0={\frac {\partial y}{\partial \lambda }}}$

${\displaystyle y=\ln(|\sec(\varphi )+\tan(\varphi )|)+C.\,}$

## 錯覺

• 澳洲：7,692,024平方公里
• 格陵蘭：2,166,086平方公里
• 日本：378,000平方公里
• 新幾內亞島：786,000平方公里

## 參考資料

• Snyder, John P. Map Projections - A Working Manual. U.S. Geological Survey Professional Paper 1395. United States Government Printing Office, Washington, D.C. 1987.可至USGS pages下载。
• Monmonier, Mark. Rhumb Lines and Map Wars. Chicago: The University of Chicago Press. 2004.
• Needham, Joseph (1986). Science and Civilization in China: Volume 3; Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books Ltd.
• Needham, Joseph (1986). Science and Civilization in China: Volume 4, Physics and Physical Technology, Part 3, Civil Engineering and Nautics. Taipei: Caves Books Ltd.