Transverse Mercator projection
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A Transverse Mercator projection is an adaptation of the Mercator projection. Both projections are cylindrical and conformal. However, in a Transverse Mercator projection, the cylinder is rotated 90° (transverse) relative to the equator so that projected surface is aligned with a ("central") meridian (or line of longitude) rather than the equator, as is the case with the regular Mercator projection.
In both the regular and transverse form of the Mercator projection, there is very little distortion of scale in the narrow region near where the projected surface is tangent, or secant, to the sphere or ellipsoid representing the Earth. The scale 5° away from the central meridian is less than 0.4% larger than the scale at the central meridian and is approximately 1.53% at an angular distance of 10°. This low level of distortion, combined with the conformal property which it inherits from the Mercator projection, make the Transverse Mercator projection ideal for mapping areas with a narrow longitudinal range, e.g., a nation such as Chile.
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[edit] Forms of the Transverse Mercator Projection
The spherical form of the Transverse Mercator projection, which uses a sphere to represent the Earth, was first presented by Johann Heinrich Lambert in 1772. An elliptical form, which uses an ellipsoidal model of the Earth, was later presented by mathematician Carl Friedrich Gauss in 1822 and was further analyzed by L. Krüger in the early 20th century. In Europe, the Transverse Mercator projection is sometimes referred to as the Gauss-Krüger or Gauss Conformal projection.
The spherical form of the Transverse Mercator projection is conformal. The distortion of scale increases entirely as a function of distance from the central meridian. The ellipsoidal form is also conformal, but scale distortion is affected to some degree by parameters of the ellipsoid and this distortion is not entirely a consistent function of distance away from the central meridian.
The projected surface can be tangent to the model of the Earth in either case, which produces a map that is true to scale along this line. The scale factor can also be reduced in order to balance out the distortion over the mapped region (this is the case in UTM, which applies a scale factor of 0.9996). In this ("secant") case, there are two lines of true scale on either side of the central meridian. These lines are parallel to the central meridian in the spherical model, but only approximately parallel in the ellipsoidal model.
The ellipsoidal form with a reduced scale factor is likely the most widely used projection in geodetic mapping, as it is employed by the U.S. Geological Survey in areas with a predominant north-south extent.
[edit] Clarification on the ellipsoidal form of Transverse Mercator
The ellipsoidal form of Transverse Mercator is characterized by the fact that it is conformal and maps the central meridian to a line with a constant (linear) scale. It is also quite difficult to work with: the latter condition implies that the vertical (or northing) coordinate on the central meridian is proportional to rectifying latitude whereas the former means that it is convenient to start with conformal latitude; since these notions of latitude do not coincide, there is no simple way to compute the Cartesian coordinates of a point on the Transverse Mercator projection from its geodetic coordinates: approximate formulas are generally found in form of convergent power series in the Earth's eccentricity and/or distance from the central meridian. Exact formulas, involving reciprocal functions and complex incomplete elliptic integrals could be given by extending to the complex domain the transformation of conformal latitude to rectifying latitude[1], but they are typically useful only for deriving the power series for numerical computations.
[edit] See also
[edit] References
Snyder, John P. (1987). Map Projections - A Working Manual. U.S. Geological Survey Professional Paper 1395. United States Government Printing Office, Washington, D.C. This paper can be downloaded from USGS pages
[edit] External links
- Table of examples and properties of all common projections, from radicalcartography.net
