Editing Mercator projection (section)

==== 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.