Transverse Mercator projection has many implementations. Louis Krüger in 1912 developed one of his two implementations[1] that expressed as a power series in the longitude difference from the central meridian. These series were recalculated by Lee in 1946,[2] by Redfearn in 1948,[3] and by Thomas in 1952.[4] They 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[5] and many others) and also international[6] mapping systems, including the Universal Transverse Mercator coordinate system (UTM).[7] [8] They are also incorporated into the Geotrans coordinate converter made available by the United States National Geospatial-Intelligence Agency.[9] When paired with a suitable geodetic datum, the series deliver high accuracy in zones less than a few degrees in east-west extent.
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
1/f
b
e
f
n
e'
\begin{align} f&=
a-b | |
a |
, e2=2f-f2,
| ||||
e' |
\\ b&=a(1-f)=a(1-e2)1/2, n=
a-b | |
a+b |
. \end{align}
\rho(\phi)
\phi
\nu(\phi)
\nu(\phi)= | a |
\sqrt{1-e2\sin2\phi |
\beta(\phi)
η(\phi)
\beta(\phi)= | \nu(\phi) |
\rho(\phi) |
, η2=\beta-1={e'2\cos2\phi}.
s=\sin\phi, c=\cos\phi, t=\tan\phi.
The article on Meridian arc describes several methods of computing
m(\phi)
\phi
\begin{align} m(\phi)&=B0\phi+B2\sin2\phi+B4\sin4\phi+B6\sin6\phi+ … , \end{align}
n3
e6
\begin{align} B0&=b(1+n+
5 | |
4 |
| ||||
n |
n3), B4=b(
15 | |
16 |
| ||||
n |
n3),\\ B2&=-b(
3 | n+ | |
2 |
3 | |
2 |
| ||||
n |
n3), B6=-b(
35 | |
48 |
n3). \end{align}
mp=m(\pi/2)=\piB0/2.
Neither the OSGB nor the UTM implementations define an inverse series for the meridian distance; instead they use an iterative scheme. For a given meridian distance
M
\phi0=M/B0
\begin{align} \phin=\phin-1+
M-m(\phin-1) | |
B0 |
, n=1,2,3,\ldots \end{align}
|M-m(\phin-1)|<0.01
The inversion can be effected by a series, presented here for later reference. For a given meridian distance,
M
\mu= | \piM |
2mp |
.
M
\begin{align} \phi&=\mu+D2\sin2\mu+D4\sin4\mu+D6\sin6\mu+D8\sin8\mu+ … ,\\ \end{align}
O(n4)
\begin{align} D2&=
3 | n- | |
2 |
27 | |
32 |
n3,& D4&=
21 | |
16 |
| ||||
n |
n4,\\[8pt] D6&=
151 | |
96 |
n3,& D8&=
1097 | |
512 |
n4. \end{align}
The normal aspect of the Mercator projection of a sphere of radius
R
x=Rλ, y=R\psi,
\psi
\begin{align} \psi&=ln\left[\tan\left(
\pi | |
4 |
+
\phi | |
2 |
\right)\right]. \end{align}
On the ellipsoid the isometric latitude becomes
\begin{align} \psi&=ln\left[\tan\left(
\pi | |
4 |
+
\phi | |
2 |
\right)\right] -
e | |
2 |
ln\left[
1+e\sin\phi | |
1-e\sin\phi |
\right]. \end{align}
\phi
λ
\psi
λ
\psi
λ
\zeta=\psi+iλ
f(\zeta)
f(\zeta)
λ=0
\begin{align} y+ix&=f(\zeta)=f(\psi+iλ)\\ &=f(\psi+i.0)+A1λ+
2+ | |
A | |
2λ |
3+ | |
A | |
3λ |
\ldots, \end{align}
f(\psi+i.0)
m(\phi)
An
f(\zeta)
m(\phi)
\psi
\phi
\psi
\phi
x
y
This section presents the eighth order series as published by Redfearn[3] (but with
x
y
λ
\omega
The direct series are developed in terms of the longitude difference from the central meridian, expressed in radians: the inverse series are developed in terms of the ratio
x/a
x
y
k
\gamma
\phi
λ
k
\gamma
x
y
In the following series
λ
λ
\phi
y
λ=0
\begin{align} x(λ,\phi)&=k0\nu\left[λc +
λ3c3W3 | + | |
3! |
λ5c5W5 | + | |
5! |
λ7c7W7 | |
7! |
\right],\\[1ex] y(λ,\phi)&=k0\left[m(\phi)+
λ2\nuc2t | + | |
2 |
| + | |||||||||||||
4! |
| + | |||||||||||||
6! |
| |||||||||||||
8! |
\right], \end{align}
The inverse series involve a further construct: the footpoint latitude. Given a point
(x,y)
(0,y)
k0
m=y/k0
\phi1
\begin{align} \mu&= | \piy |
2mpk0 |
,\\ \phi1&=\mu+D2\sin2\mu+D4\sin4\mu+D6\sin6\mu+D8\sin8\mu+ … ,\\ \end{align}
\phi1
\begin{align} λ(x,y) &=
x | - | |
c1(k0\nu1) |
x3V3 | - | |||||||||||
|
x5V5 | - | |||||||||||
|
x7V7 | ||||||||||||
|
, \\ \phi(x,y)&=\phi | - | ||||||||||||||||
|
x4\beta1t1U4 | - | |||||||||||
|
x6\beta1t1U6 | - | |||||||||||
|
x8\beta1t1U8 | ||||||||||||
|
. \end{align}
The point scale
k
k
k0
λ=0
x=0
\gamma
\begin{align}
k(λ,\phi) &=k | + | ||||||||||||||||
|
| + | |||||||||||||
24 |
| |||||||||||||
720 |
\right],\\ \gamma(λ,\phi) &=λs +
λ3c3tH3 | + | |
3 |
λ5c5tH5 | + | |
15 |
| |||||||||||||
315 |
,\\ k(x,y)&=k | + | ||||||||||||||||||||||
|
| + | |||||||||||
|
| ||||||||||||
|
\right]\\ \gamma(x,y)&=
xt1 | |
k0\nu1 |
+
| ||||||||||||
|
+
| ||||||||||||
|
+
| ||||||||||||
|
. \end{align}
\begin{align} W3&=\beta-t2\\ W5&=4\beta3(1-6t2)+\beta2(1+8t2)-2\betat2+t4\\ W7&=61-479t2+179t4-t6+O(e2)\\ W4&=4\beta2+\beta-t2\\ W6&=8\beta4(11{-}24t2)-28\beta3(1{-}6t2)+\beta2(1{-}32t2) -2\betat2+t4\\ W8&=1385-3111t2+543t4-t6+O(e2)\\ V3&=\beta1+2t
2 | |
1 |
\\ V5&=
2) | |
4\beta | |
1 |
-72\beta1
2 | |
t | |
1 |
4\\ | |
-24t | |
1 |
V7&=
6 | |
61+662t | |
1 |
\\ U4&=
2-9\beta | |
4\beta | |
1(1-t |
2\\ | |
1 |
U6&=
2) | |
8\beta | |
1 |
4) | |
+15\beta | |
1 |
\\ & +180\beta1(5t
4\\ | |
1 |
U8&=-1385-3633t
6\\ H | |
2&= |
\beta\\ H4&=4\beta3(1-6t2)+\beta2(1+24t2)-4\beta
2+16t | |
t | |
6&=61-148t |
4(11-24t | |
5&=\beta |
2)-\beta3(11-36t2) +\beta2(2-14t2)+\beta
2+2t | |
t | |
7&=17-26t |
4\\ K | |
2&=\beta |
1\\ K4&=4\beta
2) | |
1 |
-24\beta1
2\\ K | |
t | |
6&=1\\ K |
3&=2\beta
2-3\beta | |
1 |
2\\ K | |
-t | |
5&=\beta |
2) | |
1 |
2)+30\beta | |
+5\beta | |
1 |
4\\ K | |
t | |
7&=-17-77t |
6 \end{align} | |
1 |
The exact solution of Lee-Thompson,[12] implemented by Karney (2011),[13] is of great value in assessing the accuracy of the truncated Redfearn series. It confirms that the truncation error of the (eighth order) Redfearn series is less than 1 mm out to a longitude difference of 3 degrees, corresponding to a distance of 334 km from the central meridian at the equator but a mere 35 km at the northern limit of an UTM zone.
The Redfearn series become much worse as the zone widens. Karney discusses Greenland as an instructive example. The long thin landmass is centred on 42W and, at its broadest point, is no more than 750 km from that meridian whilst the span in longitude reaches almost 50 degrees. The Redfearn series attain a maximum error of 1 kilometre.
The implementations give below are examples of the use of the Redfearn series. The defining documents in various countries differ slightly in notation and, more importantly, in the neglect of some of the small terms. The analysis of small terms depends on the latitude and longitude ranges in the various grids. There are also slight differences in the formulae utilised for meridian distance: one extra term is sometimes added to the formula specified above but such a term is less than 0.1mm.
The implementation of the transverse Mercator projection in Great Britain is fully described in the OSGB document A guide to coordinate systems in Great Britain, Appendices A.1, A.2 and C.[11]
datum: OSGB36
ellipsoid: Airy 1830
major axis: 6 377 563.396
minor axis: 6 356 256.909
central meridian longitude: 2°W
central meridian scale factor : 0.9996012717
projection origin: 2°W and 0°N
true grid origin: 2°W and 49°N
false easting of true grid origin, E0 (metres): 400,000
false northing of true grid origin, N0 (metres): -100,000
E = E0 + x = 400000 + x
N = N0 + y -k0*m(49°)= y - 5527063The extent of the grid is 300 km to the east and 400 km to the west of the central meridian and 1300 km north from the false origin, (OSGB[11] Section 7.1), but with the exclusion of parts of Northern Ireland, Eire and France. A grid reference is denoted by the pair (E,N) where E ranges from slightly over zero to 800000m and N ranges from zero to 1300000m. To reduce the number of figures needed to give a grid reference, the grid is divided into 100 km squares, which each have a two-letter code. National Grid positions can be given with this code followed by an easting and a northing both in the range 0 and 99999m.
The projection formulae differ slightly from the Redfearn formulae presented here. They have been simplified by neglect of most terms of seventh and eighth order in
λ
x/a
λ
x/a
\beta
η
The OSGB manual[11] includes a discussion of the Helmert transformations which are required to link geodetic coordinates on Airy 1830 ellipsoid and on WGS84.
The article on the Universal Transverse Mercator projection gives a general survey, but the full specification is defined in U.S. Defense Mapping Agency Technical Manuals TM8358.1 and TM8358.2. This section provides details for zone 30 as another example of the Redfearn formulae (usually termed Thomas formulae in the United States.)
ellipsoid: International 1924 (a.k.a. Hayford 1909)
major axis: 6 378 388.000
minor axis: 6 356 911.946
central meridian longitude: 3°W
projection origin: 3°W and 0°N
central meridian scale factor: 0.9996
true grid origin: 3°W and 0°N
false easting of true grid origin, E0: 500,000
E = E0 + x = 500000 + x
northern hemisphere false northing of true grid origin N0: 0
northern hemisphere: N = N0 + y = y
southern hemisphere false northing of true grid origin N0: 10,000,000
southern hemisphere: N = N0 + y = 10,000,000 + yThe series adopted for the meridian distance incorporates terms of fifth order in
n
e'
n
The transverse Mercator projection in Eire and Northern Ireland (an international implementation spanning one country and part of another) is currently implemented in two ways:
datum: Ireland 1965
ellipsoid: Airy 1830 modified
major axis: 6 377 340.189
minor axis: 6 356 034.447
central meridian scale factor: 1.000035
true origin: 8°W and 53.5°N
false easting of true grid origin, E0: 200,000
false northing of true grid origin, N0: 250,000The Irish grid uses the OSGB projection formulae.
datum: Ireland 1965
ellipsoid: GRS80
major axis: 6 378 137
minor axis: 6 356 752.314140
central meridian scale factor: 0.999820
true origin: 8°W and 53.5°N
false easting of true grid origin, E0: 600,000
false northing of true grid origin, N0: 750,000This is an interesting example of the transition between use of a traditional ellipsoid and a modern global ellipsoid. The adoption of radically different false origins helps to prevent confusion between the two systems.