Editing Scale (map) (section)

===Point scale for normal cylindrical projections of the sphere===
[[File:201globe.svg|200px|right]]
The key to a ''quantitative'' understanding of scale is to consider an infinitesimal element on the sphere. The figure shows a point P at latitude <math>\varphi</math> and longitude <math>\lambda</math> on the sphere. The point Q is at latitude <math>\varphi+\delta\varphi</math> and longitude <math>\lambda+\delta\lambda</math>. The lines PK and MQ are [[meridian arc|arcs of meridians]] of length <math>a\,\delta\varphi</math> where <math>a</math> is the radius of the sphere and <math>\varphi</math> is in radian measure. The lines PM and KQ are arcs of parallel circles of length <math>(a\cos\varphi)\delta\lambda</math> with<math>\lambda</math> in radian measure. In deriving a ''point'' property of the projection ''at'' P it suffices to take an infinitesimal element PMQK of the surface: in the limit of Q approaching P such an element tends to an infinitesimally small planar rectangle.

[[File:205infinitesimal elements on sphere and plane(6).svg|thumb|350px|right|Infinitesimal elements on the sphere and a normal cylindrical projection]]
Normal cylindrical projections of the sphere have  <math>x=a\lambda</math> and <math>y</math> equal to a function of latitude only. Therefore, the infinitesimal element PMQK on the sphere projects to an infinitesimal element P'M'Q'K' which is an ''exact'' rectangle with a base <math>\delta x=a\,\delta\lambda</math> and height&nbsp;<math>\delta y</math>. By comparing the elements on sphere and projection we can immediately deduce expressions for the scale factors on parallels and meridians. (The treatment of scale in a general direction may be found [[#Mathematical addendum|below]].)
:: parallel scale factor&nbsp;&nbsp; <math>\quad k\;=\;\dfrac{\delta x}{a\cos\varphi\,\delta\lambda\,}=\,\sec\varphi\qquad\qquad{}</math>
::meridian scale factor&nbsp;  <math>\quad h\;=\;\dfrac{\delta y}{a\,\delta\varphi\,} = \dfrac{y'(\varphi)}{a}</math>
Note that the parallel scale factor <math>k=\sec\varphi</math> 
is independent of the definition of <math>y(\varphi)</math> so it is the same for all normal cylindrical projections. It is useful to note that
::at latitude 30 degrees the parallel scale is <math>k=\sec30^{\circ}=2/\sqrt{3}=1.15</math>
::at latitude 45 degrees the parallel scale is <math>k=\sec45^{\circ}=\sqrt{2}=1.414</math>
::at latitude 60 degrees the parallel scale is <math>k=\sec60^{\circ}=2</math>
::at latitude 80 degrees the parallel scale is <math>k=\sec80^{\circ}=5.76</math>
::at latitude 85 degrees the parallel scale is <math>k=\sec85^{\circ}=11.5</math>

The following examples illustrate three normal cylindrical projections and in each case the variation of scale with position and direction is illustrated by the use of [[Tissot's indicatrix]].