Editing Transverse Mercator projection (section)

===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.