Editing Transverse Mercator projection (section)

==Formulae for the spherical transverse Mercator==

===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.

===Normal and transverse graticules===
[[File:Transverse mercator graticules.svg|thumb|400px|center|Transverse mercator graticules]]
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The figure on the left shows how a transverse cylinder is related to the conventional graticule on the sphere. It is tangential to some arbitrarily chosen meridian and its axis is perpendicular to that of the sphere. The ''x''- and ''y''-axes defined on the figure are related to the equator and central meridian exactly as they are for the normal projection. In the figure on the right a rotated graticule is related to the transverse cylinder in the same way that the normal cylinder is related to the standard graticule. The 'equator', 'poles' (E and W) and 'meridians' of the rotated graticule are identified with the chosen central meridian, points on the equator 90 degrees east and west of the central meridian, and great circles through those points.

[[File:Transverse mercator geometry.svg|thumb|Transverse mercator geometry]]
The position of an arbitrary point (''φ'',''λ'') on the standard graticule can also be identified in terms of angles on the rotated graticule: ''φ′'' (angle M′CP) is an effective latitude and −''λ′'' (angle M′CO) becomes an effective longitude. (The minus sign is necessary so that (''φ′'',''λ′'') are related to the rotated graticule in the same way that (''φ'',''λ'') are related to the standard graticule). The Cartesian (''x′'',''y′'') axes are related to the rotated graticule in the same way that the axes (''x'',''y'') axes are related to the standard graticule.

The tangent transverse Mercator projection defines the coordinates (''x′'',''y′'') in terms of −''λ′'' and ''φ′'' by the transformation formulae of the tangent Normal Mercator projection:
:<math>x' = -a\lambda'\,\qquad
y'  = \frac{a}{2}
         \ln\left[\frac{1+\sin\varphi'}{1-\sin\varphi'}\right].
</math>
This transformation projects the central meridian to a straight line of finite length and at the same time projects the great circles through E and W (which include the equator) to infinite straight lines perpendicular to the central meridian. The true parallels and meridians (other than equator and central meridian) have no simple relation to the rotated graticule and they project to complicated curves.

===The relation between the graticules===
The angles of the two graticules are related by using [[spherical trigonometry]] on the spherical triangle NM′P defined by the true meridian through the origin, OM′N, the true meridian through an arbitrary point, MPN, and the great circle WM′PE. The results are:<ref name=merc/>
:<math>
\begin{align}
\sin\varphi'&=\sin\lambda\cos\varphi,\\
\tan\lambda'&=\sec\lambda\tan\varphi.
\end{align}
</math>

===Direct transformation formulae===
The direct formulae giving the Cartesian coordinates (''x'',''y'') follow immediately from the above. Setting ''x''&nbsp;=&nbsp;''y′'' and ''y''&nbsp;=&nbsp;−''x′'' (and restoring factors of ''k''{{sub|0}} to accommodate secant versions)
:<math>
\begin{align}
x(\lambda,\varphi)&= \frac{1}{2}k_0a
         \ln\left[
       \frac{1+\sin\lambda\cos\varphi}
        {1-\sin\lambda\cos\varphi}\right],\\[5px]
y(\lambda,\varphi)&= k_0 a\arctan\left[\sec\lambda\tan\varphi\right],
\end{align}
</math>
The above expressions are given in Lambert<ref name=lambert/> and also (without derivations) in Snyder,<ref name=snyder>{{cite book | author=Snyder, John P. | title=Map Projections—A Working Manual. U.S. Geological Survey Professional Paper 1395 | publisher =United States Government Printing Office, Washington, D.C. | year=1987}}This paper can be downloaded from [https://pubs.er.usgs.gov/pubs/pp/pp1395 USGS pages.] It gives full details of most projections, together with interesting introductory sections, but it does not derive any of the projections from first principles.</ref> Maling<ref name=maling>{{cite book | author=Maling, Derek Hylton | title=Coordinate Systems and Map Projections | publisher =Pergamon Press| year=1992|isbn=978-0-08-037233-4 |edition=second}}.</ref> and Osborne<ref name=merc>[https://web.archive.org/web/20130924093049/http://www.mercator99.webspace.virginmedia.com/ The Mercator Projections] Detailed derivations of all formulae quoted in this article</ref> (with full details).

===Inverse transformation formulae===
Inverting the above equations gives
:<math>
\begin{align}
  \lambda(x,y)&
= \arctan\left[ \sinh\frac{x}{k_0a}
                     \sec\frac{y}{k_0a} \right],
\\[5px]
  \varphi(x,y)&= \arcsin\left[ \mbox{sech}\;\frac{x}{k_0a}
                        \sin\frac{y}{k_0a} \right].
\end{align}
</math>

===Point scale===
In terms of the coordinates with respect to the rotated graticule the [[Scale (map)|point scale]] factor is given by ''k''&nbsp;=&nbsp;sec&nbsp;''φ′'': this may be expressed either in terms of the geographical coordinates or in terms of the projection coordinates:
:<math>
\begin{align}
  k(\lambda,\varphi)&=\frac{k_0}{\sqrt{1-\sin^2\lambda\cos^2\varphi}},\\[5px]
k(x,y)&=k_0\cosh\left(\frac{x}{k_0a}\right).
\end{align}
</math>
The second expression shows that the scale factor is simply a function of the distance from the central meridian of the projection. A typical value of the scale factor is ''k''{{sub|0}}&nbsp;=&nbsp;0.9996 so that ''k''&nbsp;=&nbsp;1 when ''x'' is approximately 180&nbsp;km. When ''x'' is approximately 255&nbsp;km and ''k''{{sub|0}}&nbsp;=&nbsp;1.0004: the scale factor is within 0.04% of unity over a strip of about 510&nbsp;km wide.

{{anchor|convergence}}

===Convergence===
[[File:Transverse mercator convergence.svg|right|thumb|upright|The angle of convergence]]

The convergence angle ''γ'' at a point on the projection is defined by the angle measured ''from'' the projected meridian, which defines true north, ''to'' a grid line of constant ''x'', defining grid north. Therefore, ''γ'' is positive in the quadrant north of the equator and east of the central meridian and also in the quadrant south of the equator and west of the central meridian. The convergence must be added to a grid bearing to obtain a bearing from true north. For the secant transverse Mercator the convergence may be expressed<ref name=merc/> either in terms of the geographical coordinates or in terms of the projection coordinates:
:<math>
\begin{align}
\gamma(\lambda,\varphi)&=\arctan(\tan\lambda\sin\varphi),\\[5px]
\gamma(x,y)&=\arctan\left(\tanh\frac{x}{k_0a}\tan\frac{y}{k_0a}\right).
\end{align}
</math>