Editing Scale (map) (section)

===Mathematical addendum===

[[File:205infinitesimal elements on sphere and plane(6).svg|thumb|350px|right|Infinitesimal elements on the sphere and a normal cylindrical projection]]

For normal cylindrical projections the geometry of the infinitesimal elements gives
::<math>
   \text{(a)}\quad
   \tan\alpha=\frac{a\cos\varphi\,\delta\lambda}{a\,\delta\varphi}, </math>
::<math>
   \text{(b)}\quad
   \tan\beta=\frac{\delta x}{\delta y}
              =\frac{a\,\delta \lambda}{\delta y}.
</math>
The relationship between the angles <math>\beta</math> and <math>\alpha</math> is
::<math>     \text{(c)}\quad
\tan\beta=\frac{a\sec\varphi}{y'(\varphi)} \tan\alpha.\,</math>
For the Mercator projection  <math>y'(\varphi)=a\sec\varphi</math> giving <math>\alpha=\beta</math>: angles are preserved. (Hardly surprising since this is the relation used to derive Mercator). For the equidistant and Lambert projections we have <math>y'(\varphi)=a</math> and <math>y'(\varphi)=a\cos\varphi</math> respectively so the relationship between <math>\alpha</math> and <math>\beta</math> depends upon the latitude&nbsp;<math>\varphi</math>. 
Denote the point scale at P when the infinitesimal element PQ makes an angle 
<math>\alpha \,</math> with the meridian by <math>\mu_{\alpha}.</math> It is given by the ratio of distances:

::<math> 
\mu_\alpha = \lim_{Q\to P}\frac{P'Q'}{PQ}
= \lim_{Q\to P}\frac{\sqrt{\delta x^2 +\delta y^2}}
{\sqrt{ a^2\,  \delta\varphi^2+a^2\cos^2 \varphi\, \delta\lambda^2}}.
</math>
Setting <math>\delta x=a\,\delta\lambda</math> and substituting <math>\delta\varphi</math> and <math>\delta y</math> from equations (a) and (b) respectively gives
::<math>\mu_\alpha(\varphi) = \sec\varphi \left[\frac{\sin\alpha}{\sin\beta}\right].</math>

For the projections other than Mercator we must first calculate <math>\beta</math> from <math>\alpha</math> and <math>\varphi</math> using equation (c), before we can find <math>\mu_{\alpha}</math>. For example, the equirectangular projection has <math>y'=a</math> so that 
::<math>\tan\beta=\sec\varphi \tan\alpha.\,</math>
If we consider a line of constant slope  <math>\beta</math> on the projection both the corresponding value of <math>\alpha</math> and the scale factor along the line are complicated functions of <math>\varphi</math>. There is no simple way of transferring a general finite separation to a bar scale and obtaining meaningful results.