Editing Scale (map) (section)

====The equirectangular projection====
[[File:Tissot indicatrix world map equirectangular proj.svg|thumb|right|350px|The equidistant projection with [[Tissot's indicatrix]] of deformation]] 
The [[equirectangular projection]],<ref name=snyder/><ref name=flattening/><ref name=merc/> also known as the ''Plate Carrée'' (French for "flat square") or (somewhat misleadingly) the equidistant projection, is defined by 
:<math>x = a\lambda,</math>&nbsp;&nbsp; <math>y = a\varphi,</math>
where <math>a</math> is the radius of the sphere, <math>\lambda</math> is the longitude from the central meridian of the projection (here taken as the Greenwich meridian at <math>\lambda =0</math>) and <math>\varphi</math> is the latitude. Note that <math>\lambda</math> and <math>\varphi</math> are in radians (obtained by multiplying the degree measure by a factor of <math>\pi</math>/180). The longitude <math>\lambda</math> is in the range <math>[-\pi,\pi]</math> and the latitude <math>\varphi</math> is in the range <math>[-\pi/2,\pi/2]</math>.

Since <math>y'(\varphi)=1</math> the previous section gives
: parallel scale,&nbsp; <math>\quad k\;=\;\dfrac{\delta x}{a\cos\varphi\,\delta\lambda\,}=\,\sec\varphi\qquad\qquad{}</math>
: meridian scale  <math>\quad h\;=\;\dfrac{\delta y}{a\,\delta\varphi\,}=\,1</math>
For the calculation of the point scale in an arbitrary direction see [[#Mathematical addendum|addendum]].

The figure illustrates the [[Tissot's indicatrix|Tissot indicatrix]] for this projection. On the equator h=k=1 and the circular elements are undistorted on projection. At higher latitudes the circles are distorted into an ellipse given by stretching in the parallel direction only: there is no distortion in the meridian direction. The ratio of the major axis to the minor axis is <math>\sec\varphi</math>. Clearly the area of the ellipse increases by the same factor.

It is instructive to consider the use of bar scales that might appear on a printed version of this projection. The scale is true (k=1) on the equator so that multiplying its length on a printed map by the inverse of the RF (or principal scale) gives the actual circumference of the Earth. The bar scale on the map is also drawn at the true scale so that transferring a separation between two points on the equator to the bar scale will give the correct distance between those points. The same is true on the meridians. On a parallel other than the equator the scale is <math>\sec\varphi</math> so when we transfer a separation from a parallel to the bar scale we must divide the bar scale distance by this factor to obtain the distance between the points when measured along the parallel (which is not the true distance along a [[great circle]]). On a line at a bearing of say 45 degrees (<math>\beta=45^{\circ}</math>) the scale is continuously varying with latitude and transferring a separation along the line to the bar scale does not give a distance related to the true distance in any simple way. (But see [[#Mathematical addendum|addendum]]). Even if a distance along this line of constant planar angle could be worked out, its relevance is questionable since such a line on the projection corresponds to a complicated curve on the sphere. For these reasons bar scales on small-scale maps must be used with extreme caution.