Editing Transverse Mercator projection (section)

===Spherical normal Mercator revisited===
[[File:Cylindrical Projection basics.svg|thumb|400px|center|The normal aspect of a tangent cylindrical projection of the sphere]]
The normal cylindrical projections are described in relation to a cylinder tangential at the equator with axis along the polar axis of the sphere. The cylindrical projections are constructed so that all points on a meridian are projected to points with <math>x = a\lambda</math> (where <math>a</math> is the [[Earth radius]]) and <math>y</math> is a prescribed function of <math>\phi</math>. For a tangent Normal Mercator projection the (unique) formulae which guarantee conformality are:<ref name=merc/>
:<math>x = a\lambda\,,\qquad
y  = a\ln \left[\tan \left(\frac{\pi}{4} + \frac{\varphi}{2} \right)\right]
   = \frac{a}{2}\ln\left[\frac{1+\sin\varphi}{1-\sin\varphi}\right].
</math>
Conformality implies that the [[Scale (map)|point scale]], ''k'', is independent of direction: it is a function of latitude only:
:<math>k(\varphi)=\sec\varphi.\,</math>
For the secant version of the projection there is a factor of ''k''{{sub|0}} on the right hand side of all these equations: this ensures that the scale is equal to ''k''{{sub|0}} on the equator.