# 麥卡托投影法

## 数学计算

\begin{align} 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{align}

\begin{align} \varphi & = 2\tan^{-1}(e^y) - \frac{\pi}{2} \\ & = \tan^{-1}(\sinh(y)) \\ \lambda & = x + \lambda_0. \end{align}

## 公式推导

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

x = λ 可知

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

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

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

## 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.