Transverse Mercator projection has many implementations. Louis Krüger in 1912 developed one of his two implementations that expressed as a power series in the longitude difference from the central meridian. These series were recalculated by Laurence Patrick Lee in 1946, by J.C.B. Redfearn in 1948, and by Paul D. Thomas in 1952. These series are often referred to as the Redfearn series, or the Thomas series. This implementation is of great importance since it is widely used in the U.S. State Plane Coordinate System, in national (Great Britain, Ireland and many others) and also international mapping systems, including the Universal Transverse Mercator coordinate system (UTM). They are also incorporated into the GEOTRANS coordinate converter made available by the United States National Geospatial-Intelligence Agency. When paired with a suitable geodetic datum, the series deliver high accuracy in zones less than a few degrees in east-west extent.
Preliminaries I: datum and ellipsoid parameters The series must be used with a geodetic datum which specifies the position, orientation and shape of a reference ellipsoid. Although the projection formulae depend only on the shape parameters of the reference ellipsoid the full set of datum parameters is necessary to link the projection coordinates to true positions in three-dimensional space. The datums and reference ellipsoids associated with particular implementations of the Redfearn formulae are listed below. A comprehensive list of important ellipsoids is given in the article on the Figure of the Earth. In specifying ellipsoids it is normal to give the semi-major axis (equatorial axis), a {\displaystyle a} , along with either the inverse flattening, 1 / f {\displaystyle 1/f} , or the semi-minor axis (polar axis), b {\displaystyle b} , or sometimes both. The series presented below use the eccentricity, e {\displaystyle e} , in preference to the flattening, f {\displaystyle f} . In addition they use the parameters n {\displaystyle n} , called the third flattening, and e ′ {\displaystyle e'} , the second eccentricity. There are only two independent shape parameters and there are many relations between them: in particular
f = a − b a , e 2 = 2 f − f 2 , e ′ 2 = e 2 1 − e 2 b = a ( 1 − f ) = a ( 1 − e 2 ) 1 / 2 , n = a − b a + b . {\displaystyle {\begin{aligned}f&={\frac {a-b}{a}},\qquad e^{2}=2f-f^{2},\qquad e'^{2}={\frac {e^{2}}{1-e^{2}}}\\b&=a(1-f)=a(1-e^{2})^{1/2},\qquad n={\frac {a-b}{a+b}}.\end{aligned}}}
The projection formulae also involve ρ ( ϕ ) {\displaystyle \rho (\phi )} , the radius of curvature of the meridian (at latitude ϕ {\displaystyle \phi } ), and ν ( ϕ ) {\displaystyle \nu (\phi )} , the radius of curvature in the prime vertical. (The prime vertical is the vertical plane orthogonal to the meridian plane at a point on the ellipsoid). The radii of curvature are defined as follows:
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