Editing Mercator projection (section)

=== Mercator projection transformations ===
==== Derivation ====

As discussed above, the isotropy condition implies that ''h'' = ''k'' = {{nowrap|sec ''φ''}}.  Consider a point on the globe of radius ''R'' with longitude ''λ'' and latitude ''φ''.  If ''φ'' is increased by an infinitesimal amount, ''dφ'', the point moves ''R'' ''dφ'' along a meridian of the globe of radius ''R'', so the corresponding change in ''y'', ''dy'', must be ''hR'' ''dφ'' = ''R''&nbsp;sec&nbsp;''φ'' ''dφ''. Therefore ''y′''(''φ'')&nbsp;=&nbsp;''R''&nbsp;sec&nbsp;''φ''.  Similarly, increasing ''λ'' by ''dλ'' moves the point ''R'' cos ''φ'' ''dλ'' along a parallel of the globe, so ''dx'' = ''kR'' cos ''φ'' ''dλ'' = ''R'' ''dλ''.  That is, ''x′''(''λ'')&nbsp;=&nbsp;''R''. Integrating the equations
:<math>x'(\lambda) = R, \qquad y'(\varphi) = R\sec\varphi,</math>

with ''x''(''λ''<sub>0</sub>)&nbsp;=&nbsp;0 and ''y''(0)&nbsp;=&nbsp;0, gives ''x(λ)'' and ''y(φ)''.  The value ''λ''<sub>0</sub> is the longitude of an arbitrary central meridian that is usually, but not always, [[prime meridian|that of Greenwich]] (i.e., zero). The angles ''λ'' and ''φ'' are expressed in radians. By the [[integral of the secant function]],<ref name=gudermannian>[[#NIST|NIST.]] See Sections [https://dlmf.nist.gov/4.26#ii 4.26#ii] and [https://dlmf.nist.gov/4.23#viii 4.23#viii]</ref><ref name=osborne>{{harvnb|Osborne|2013|loc=Chapter 2}}</ref>
[[File:Mercator y plot.svg|right]]
:<math>
x = R( \lambda - \lambda_0), \qquad
y = R\ln \left[\tan \left(\frac{\pi}{4} + \frac{\varphi}{2} \right) \right].
</math>

The function ''y''(''φ'') is plotted alongside ''φ'' for the case ''R''&nbsp;=&nbsp;1: it tends to infinity at the poles. The linear ''y''-axis values are not usually shown on printed maps; instead some maps show the non-linear scale of latitude values on the right. More often than not the maps show only a graticule of selected meridians and parallels.

==== Inverse transformations ====
:<math>
\lambda = \lambda_0 + \frac{x}{R}, \qquad
\varphi = 2\tan^{-1}\left[\exp\left(\frac{y}{R}\right)\right] - \frac{\pi}{2} \,.
</math>

The expression on the right of the second equation defines the [[Gudermannian function]]; i.e., ''φ''&nbsp;=&nbsp;gd({{sfrac|''y''|''R''}}): the direct equation may therefore be written as ''y''&nbsp;=&nbsp;''R''·gd<sup>−1</sup>(''φ'').<ref name=gudermannian/>

==== Alternative expressions ====
There are many alternative expressions for ''y''(''φ''), all derived by elementary manipulations.<ref name=osborne/>
:<math>
\begin{align}
y & = &  \frac {R}{2} \ln \left[ \frac {1 + \sin\varphi}{1 - \sin\varphi} \right]
& = &  {R} \ln \left[ \frac {1 + \sin\varphi}{\cos\varphi} \right]
& =    R\ln \left(\sec\varphi + \tan\varphi\right) \\[2ex]
& = &  R\tanh^{-1}\left(\sin\varphi\right)
& = &  R\sinh^{-1}\left(\tan\varphi\right)
& =    R\operatorname{sgn}(\varphi)\cosh^{-1}\left(\sec\varphi\right)
=    R\operatorname{gd}^{-1}(\varphi) .
\end{align}
</math>

Corresponding inverses are:
:<math>
\varphi = \sin^{-1}\left(\tanh\frac{y}{R}\right)
= \tan^{-1}\left(\sinh\frac{y}{R}\right)
= \operatorname{sgn}(y)\sec^{-1}\left(\cosh\frac{y}{R}\right)
= \operatorname{gd}\frac{y}{R}.
</math>
For angles expressed in degrees:
:<math>
x = \frac{\pi R(\lambda^\circ-\lambda^\circ_0)}{180}, \qquad\quad
y = R\ln \left[\tan \left(45 + \frac{\varphi^\circ}{2} \right) \right].
</math>

The above formulae are written in terms of the globe radius ''R''. It is often convenient to work directly with the map width ''W''&nbsp;=&nbsp;2{{pi}}''R''. For example, the basic transformation equations become
:<math>
x = \frac{W}{2\pi}\left( \lambda - \lambda_0\right), \qquad\quad
y = \frac{W}{2\pi}\ln \left[\tan \left(\frac{\pi}{4} + \frac{\varphi}{2} \right) \right].
</math>

==== Truncation and aspect ratio ====
The ordinate ''y'' of the Mercator projection becomes infinite at the poles and the map must be truncated at some latitude less than ninety degrees. This need not be done symmetrically. Mercator's original map is truncated at 80°N and 66°S with the result that European countries were moved toward the centre of the map. The [[aspect ratio (image)|aspect ratio]] of his map is {{sfrac|198|120}} = 1.65. Even more extreme truncations have been used: a [[:File:Pieni 2 0791.jpg|Finnish school atlas]] was truncated at approximately 76°N and 56°S, an aspect ratio of 1.97.

Much Web-based mapping uses a zoomable version of the Mercator projection with an aspect ratio of one. In this case the maximum latitude attained must correspond to ''y''&nbsp;=&nbsp;±{{sfrac|''W''|2}}, or equivalently {{sfrac|''y''|''R''}}&nbsp;=&nbsp;{{pi}}. Any of the inverse transformation formulae may be used to calculate the corresponding latitudes:
:<math>
\varphi = \tan^{-1}\left[\sinh\left(\frac{y}{R}\right)\right]
= \tan^{-1}\left[\sinh\pi\right]
= \tan^{-1}\left[11.5487\right]
= 85.05113^\circ.
</math>