Editing Scale (map) (section)

===Secant, or modified, projections===
{{comparison_of_cartography_surface_development.svg|300px}}

The basic idea of a secant projection is that the sphere is projected to a cylinder which intersects the sphere at two parallels, say <math>\varphi_1</math> north and south. Clearly the scale is now true at these latitudes whereas parallels beneath these latitudes are contracted by the projection and their (parallel) scale factor must be less than one. The result is that deviation of the scale from unity is reduced over a wider range of latitudes.

[[File:Cylindrical Projection secant.svg|thumb|center|400px]]

As an example, one possible secant Mercator projection is defined by 
:<math>x = 0.9996a\lambda \qquad\qquad y  = 0.9996a\ln \left(\tan \left(\frac{\pi}{4} + \frac{\varphi}{2} \right) \right).</math>
The numeric multipliers do not alter the shape of the projection but it does mean that the scale factors are modified:
::: secant Mercator scale, &nbsp;&nbsp;<math>\quad k\;=0.9996\sec\varphi.</math>
Thus 
* the scale on the equator is 0.9996,
* the scale is ''k''&nbsp;=&nbsp;1 at a latitude given by  <math>\varphi_1</math> where   <math>\sec\varphi_1=1/0.9996=1.00004</math> so that <math>\varphi_1=1.62</math> degrees,
*k=1.0004 at a latitude <math>\varphi_2</math> given by <math>\sec\varphi_2=1.0004/0.9996=1.0008</math> for which <math>\varphi_2=2.29</math> degrees. Therefore, the projection has <math>1<k<1.0004</math>, that is an accuracy of 0.04%, over a wider strip of 4.58 degrees (compared with 3.24 degrees for the tangent form).
This is illustrated by the lower (green) curve in the figure of the previous section.

Such narrow zones of high accuracy are used in the UTM and the British OSGB projection, both of which are secant, transverse Mercator on the ellipsoid with the scale on the central meridian constant at <math>k_0=0.9996</math>. The isoscale lines with <math>k=1</math> are slightly curved lines approximately 180&nbsp;km east and west of the central meridian. The maximum value of the scale factor is 1.001 for UTM and 1.0007 for OSGB.

The lines of unit scale at latitude <math>\varphi_1</math> (north and south), where the cylindrical projection surface intersects the sphere, are the '''standard parallels''' of the secant projection.

Whilst a narrow band with <math>|k-1|<0.0004</math> is important for high accuracy mapping at a large scale, for world maps much wider spaced standard parallels are used to control the scale variation. Examples are 
*Behrmann with standard parallels at 30N, 30S.
*Gall equal area with standard parallels at 45N, 45S.

[[File:cyl proj scale Lambert Gall.svg|thumb|right|350px|Scale variation for the Lambert (green) and Gall (red) equal area projections.]]
The scale plots for the latter are shown below compared with the Lambert equal area scale factors. In the latter the equator is a single standard parallel and the parallel scale increases from k=1 to compensate the decrease in the meridian scale. For the Gall the parallel scale is reduced at the equator (to k=0.707) whilst the meridian scale is increased (to k=1.414). This gives rise to the gross distortion of shape in the Gall-Peters projection. (On the globe Africa is about as long as it is broad). Note that the meridian and parallel scales are both unity on the standard parallels.