Editing Scale (map) (section)

====Mercator projection====

[[File:Tissot indicatrix world map Mercator proj.svg|thumb|right|350px|The Mercator projection with [[Tissot's indicatrix]] of deformation. (The distortion increases without limit at higher latitudes)]]

The [[Mercator projection]] maps the sphere to a rectangle (of infinite extent in the  <math>y</math>-direction) by the equations<ref name=snyder/><ref name=flattening/><ref name=merc/> 
:<math>x = a\lambda\,</math>
:<math>y  = a\ln \left[\tan \left(\frac{\pi}{4} + \frac{\varphi}{2} \right) \right]</math>
where a, <math>\lambda\,</math> and  <math>\varphi \,</math> are as in the previous example. Since <math>y'(\varphi)=a\sec\varphi</math> the scale factors are:
:parallel scale &nbsp;&nbsp;&nbsp;&nbsp;<math>k\;=\;\dfrac{\delta x}{a\cos\varphi\,\delta\lambda\,}=\,\sec\varphi.</math>
:meridian scale &nbsp;&nbsp;<math>h\;=\;\dfrac{\delta y}{a\,\delta\varphi\,}=\,\sec\varphi.</math>

In the mathematical [[#Mathematical addendum|addendum]] it is shown that the point scale in an arbitrary direction is also equal to <math>\sec\varphi</math> so the scale is isotropic (same in all directions), its magnitude increasing with latitude as <math>\sec\varphi</math>. In the Tissot diagram each infinitesimal circular element preserves its shape but is enlarged more and more as the latitude increases.