Editing Transverse Mercator projection (section)

==Spherical transverse Mercator==
In constructing a map on any projection, a [[Spherical Earth|sphere]] is normally chosen to model the Earth when the extent of the mapped region exceeds a few hundred kilometers in length in both dimensions. For maps of smaller regions, an [[Figure of the Earth|ellipsoidal model]] must be chosen if greater accuracy is required; see next section. The spherical form of the transverse Mercator projection was one of the seven new projections presented, in 1772, by [[Johann Heinrich Lambert]].<ref name=lambert >Lambert, Johann Heinrich. 1772. ''Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten''. In [http://www.kuttaka.org/~JHL/L1772a.pdf Beyträge zum Gebrauche der Mathematik und deren Anwendung, part 3], section 6)</ref><ref name=wangerin>Albert Wangerin
(Editor), 1894. ''[[Ostwalds Klassiker der exakten Wissenschaften]]''
('''54'''). Published by Wilhelm Engelmann. This is Lambert's paper with additional comments by the editor. Available at the [http://name.umdl.umich.edu/ABR2581.0001.001 University of Michigan Historical Math Library].</ref> (The text is also available in a modern English translation.<ref name="tobler">{{Cite web | last=Tobler | first=Waldo R. | title=Notes and Comments on the Composition of Terrestrial and Celestial Maps | year=1972 | publisher=University of Michigan Press| url=http://store.esri.com/esri/showdetl.cfm?SID=2&Product_ID=1284&Category_ID=38 | archive-url=https://web.archive.org/web/20160304023959/http://store.esri.com/esri/showdetl.cfm?SID=2&Product_ID=1284&Category_ID=38 | archive-date=2016-03-04}}</ref>) Lambert did not name his projections; the name ''transverse Mercator'' dates from the second half of the nineteenth century.<ref name=flattening>{{cite book | author=Snyder, John P. | title=Flattening the Earth: Two Thousand Years of Map Projections | publisher =University of Chicago Press| year=1993|isbn=978-0-226-76747-5 | page=82}} This is an excellent survey of virtually all known projections from antiquity to 1993.</ref> The principal properties of the transverse projection are here presented in comparison with the properties of the normal projection.

===Normal and transverse spherical projections===
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{| style="text-align:left" style="margin: 1em auto 1em auto"
|-valign=top
! width="1%" |
! width="36%"|Normal Mercator
! width="3%"|
! width="1%" |
! width="36%" |Transverse Mercator
|-valign=top
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| align="center" width="300px" | [[File:MercNormSph.png|center|thumb|upright|Spherical Normal (equatorial) Mercator (truncated at ''y''&nbsp;=&nbsp;±π, corresponding to approximately 85 degrees).]]
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| align="center" width="300px" | [[File:MercTranSph.png|center|thumb|upright|Spherical transverse Mercator (truncated at ''x''&nbsp;=&nbsp;±''π'' in units of Earth radius).]]
|-valign=top
| •
|The central meridian projects to the straight line ''x''&nbsp;=&nbsp;0. Other meridians project to straight lines with ''x''&nbsp;constant.
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| •
|The central meridian projects to the straight line ''x''&nbsp;=&nbsp;0. Meridians 90 degrees east and west of the central meridian project to lines of constant&nbsp;''y'' through the projected poles. All other meridians project to complicated curves.
|-valign=top
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| The equator projects to the straight line ''y''&nbsp;=&nbsp;0 and parallel circles project to straight lines of constant&nbsp;''y''.
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|The equator projects to the straight line ''y''&nbsp;=&nbsp;0 but all other parallels are complicated closed curves.
|-valign=top
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|Projected meridians and parallels intersect at right angles.
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|Projected meridians and parallels intersect at right angles.
|-valign=top
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|The projection is unbounded in the ''y'' direction. The poles lie at infinity.
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|The projection is unbounded in the ''x'' direction. The points on the equator at ninety degrees from the central meridian are projected to infinity.
|-valign=top
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|The projection is conformal. The shapes of small elements are well preserved.
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|The projection is conformal. The shapes of small elements are well preserved.
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|Distortion increases with&nbsp;''y''. The projection is not suited for world maps. Distortion is small near the equator and the projection (particularly in its ellipsoidal form) is suitable for accurate mapping of equatorial regions.
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|Distortion increases with&nbsp;''x''. The projection is not suited for world maps. Distortion is small near the central meridian and the projection (particularly in its ellipsoidal form) is suitable for accurate mapping of narrow regions.
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|Greenland is almost as large as Africa; the actual area is about one fourteenth that of Africa.
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|When Greenland and Africa are both near the central meridian, their shapes are good and the ratio of the areas is a good approximation to actual values.
|-valign=top
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|The [[Scale (map)|point scale factor]] is independent of direction. It is a function of&nbsp;''y'' on the projection. (On the sphere it depends on latitude only.) The scale is true on the equator.
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|The point scale factor is independent of direction. It is a function of ''x'' on the projection. (On the sphere it depends on both latitude and longitude.) The scale is true on the central meridian.
|-valign=top
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|The projection is reasonably accurate near the equator. Scale at an angular distance of 5° (in latitude) away from the equator is less than 0.4% greater than scale at the equator, and is about 1.54% greater at an angular distance of&nbsp;10°.
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| •
|The projection is reasonably accurate near the central meridian. Scale at an angular distance of 5° (in longitude) away from the central meridian is less than 0.4% greater than scale at the central meridian, and is about 1.54% at an angular distance of&nbsp;10°.
|-valign=top
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|In the secant version the scale is reduced on the equator and it is true on two lines parallel to the projected equator (and corresponding to two parallel circles on the sphere).
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|In the secant version the scale is reduced on the central meridian and it is true on two lines parallel to the projected central meridian. (The two lines are not meridians.)
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|Convergence (the angle between projected meridians and grid lines with ''x'' constant) is identically zero. Grid north and true north coincide.
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|Convergence is zero on the equator and non-zero everywhere else. It increases as the poles are approached. Grid north and true north do not coincide.
|-valign=top
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|[[Rhumb line]]s (of constant azimuth on the sphere) project to straight lines.
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|right
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===[[Tissot's indicatrix|Tissot's Indicatrix]] in transverse projections===
[[File:Mercator P T Tissot.gif|thumb|500px|right|Continuous transformation from the normal [[Mercator projection|Mercator projection]] to the transverse version, with the central meridian along the x-axis. The [[Tissot's indicatrix|Tissot indicatrices]] show that conformality is preserved.]]

The [[Mercator projection]] is a conformal transformation, independently of whether it is normal, oblique or transverse. This property is shown with an animation passing smoothly from the normal aspect (equatorial projection) to the transverse aspect of a spherical Mercator projection. The indicatrices remain circular everywhere on the map, though of different sizes as a result of the local scale changes. 

During the transformation the central meridian is rotated from the y-axis in the polar projection to the x-axis in the transverse projection, as though the Earth were rotated inside the ''projection cylinder''. In both cases cropping is applied symmetrically on the y-axis, the cylinder axis, such that the map’s outline does not change. The equator of the Earth is highlighted as a thick green line. It extends to infinity in the transverse projection.