# Transverse Mercator

Name | Transverse Mercator
| Gauss-Kruger
EPSG Code | 9807
GeoTIFF Code | CT_TransverseMercator (1)
OGC WKT Name | Transverse_Mercator
Supported By | EPSG, GeoTIFF, PROJ.4, OGC WKT

### Projection Parameters

Name | EPSG # | GeoTIFF ID | OGC WKT | Units | Notes
--- | --- | --- | --- | --- | ---
Latitude of natural origin | 1 | NatOriginLat | latitude_of_origin | Angular |
Longitude of natural origin | 2 | NatOriginLong | central_meridian | Angular |
Scale factor at natural origin | 5 | ScaleAtNatOrigin | scale_factor | Unitless |
False Easting | 6 | FalseEasting | false_easting | Linear |
False Northing | 7 | FalseNorthing | false_northing | Linear |

### PROJ.4 Organization

      +proj=tmerc +lat_0=Latitude of natural origin
                  +lon_0=Longitude of natural origin
                  +k=Scale factor at natural origin
                  +x_0=False Easting
                  +y_0=False Northing

### EPSG Notes

The Transverse Mercator projection in its various forms is the most widely used projected coordinate system for world topographical and offshore mapping. All versions have the same basic characteristics and formulas. The differences which distinguish the different forms of the projection which are applied in different countries arise from variations in the choice of the coordinate transformation parameters, namely the latitude of the origin, the longitude of the origin (central meridian), the scale factor at the origin (on the central meridian), and the values of False Easting and False Northing, which embody the units of measurement, given to the origin. Additionally there are variations in the width of the longitudinal zones for the projections used in different territories. The following table indicates the variations in the projection parameters which distinguish the different forms of the Transverse Mercator projection and are used in the Epicentre Transverse Mercator projection method:

Name | Areas used | Central meridian(s) | Latitude of origin | CM Scale Factor | Zone width | False Easting at origin | False Northing at origin
--- | --- | --- | --- | --- | --- | --- | ---
Transverse Mercator | Various, world wide | Various | Various | Various | Usually less than 6* | Various | Various
Transverse Mercator south oriented | South Africa | 2* intervals E of 11*E | 0* | 1.000000 | 2* | 0m | 0m
UTM North hemisphere | World wide | 6* intervals* E & W of 3* E & W | Always 0* | Always 0.9996 | Always 6* | 500000m | 0m
UTM South hemisphere | World wide | 6* intervals E & W of 3* E & W | Always 0* | Always 0.9996 | Always 6* | 500000m | 10000000m
Gauss-Kruger | Former USSR Yugoslavia, Germany, S. America | Various, according to area of cover | Usually 0* | Usually 1.000000 | Usually less than 6*, often less than 4* | Various but often 500000 prefixed by zone number | Various
Gauss Boaga | Italy | Various | Various | 0.9996 | 6* | Various | 0m

The most familiar and commonly used Transverse Mercator is the Universal Transverse Mercator (UTM) whose natural origin lies on the equator. However, some territories use a Transverse Mercator with a natural origin at a latitude closer to that territory. In Epicentre the coordinate transformation method is the same for all forms of the Transverse Mercator projection. The formulas to derive the projected Easting and Northing coordinates are in the form of a series as follows:

    	Easting, E =  FE + k0*[A + (1 - T + C)A3/6 + (5 - 18T + T2 + 72C - 58e'2)A5/120]

    	Northing, N =  FN + k0{M - M0 + *tan*[A2/2 + (5 - T + 9C + 4C2)A4/24 +
    				(61 - 58T + T2 + 600C - 330e'2)A6/720]}
    where	T = tan2*
    	C = e2 cos2*/(1 - e2) = e'2 cos2*
    	A = (* - *0)cos*, with * and *0 in radians
    	M = a[(1 - e2/4 - 3e4/64 - 5e6/256 -....)* - (3e2/8 + 3e4/32 + 45e6/1024+....)sin2*
    		+ (15e4/256 + 45e6/1024 +.....)sin4* - (35e6/3072 + ....)sin6* + .....]
    	with * in radians and M0 for *0, the latitude of the origin, derived in the same way.

The reverse formulas to convert Easting and Northing projected coordinates to latitude and longitude are:

    	* = *1 - (*1tan*1/*1)[D2/2 - (5 + 3T1 + 10C1 - 4C12 - 9e'2)D4/24
    			+ (61 + 90T1 + 298C1 + 45T12 - 252e'2 - 3C12)D6/720]
    	* = *0 + [D - (1 + 2T1 + C1)D3/6 + (5 - 2C1 + 28T1 - 3C12 + 8e'2
    			+ 24T12)D5/120] / cos*1
    where *1 may be found as for the Cassini projection from:

    	*1 = *1 + (3e1/2 - 27e13/32 +.....)sin2*1 + (21e12/16 -55e14/32 + ....)sin4*1
    		+ (151e13/96 +.....)sin6*1 + (1097e14/512 - ....)sin8*1 + ......
    and where
    	e1 = [1- (1 - e2)1/2]/[1 + (1 - e2)1/2]
    	*1 = M1/[a(1 - e2/4 - 3e4/64 - 5e6/256 - ....)]
    	M1 = M0 + (N - FN)/k0
    	T1 = tan2*1
    	C1 = e'2cos2*1
    	D = (E - FE)/(*1k0), with *1 = * for *1

For areas south of the equator the value of latitude * will be negative and the formulas above, to compute the E and N, will automatically result in the correct values. Note that the false northings of the origin, if the equator, will need to be large to avoid negative northings and for the UTM projection is in fact 10,000,000m. Alternatively, as in the case of Argentina's Transverse Mercator (Gauss-Kruger) zones, the origin is at the south pole with a northings of zero. However each zone central meridian takes a false easting of 500000m prefixed by an identifying zone number. This ensures that instead of points in different zones having the same eastings, every point in the country, irrespective of its projection zone, will have a unique set of projected system coordinates. Strict application of the above formulas, with south latitudes negative, will result in the derivation of the correct Eastings and Northings. Similarly, in applying the reverse formulas to determine a latitude south of the equator, a negative sign for * results from a negative *1 which in turn results from a negative M1.","For Projected Coordinate System OSGB 1936 / British National Grid

    Parameters:
    Ellipsoid  Airy 1830  a = 6377563.396 m  1/f = 299.32496
    then e'^2 = 0.00671534 and e^2 = 0.00667054

    Latitude Natural Origin         49o00'00""N   = 0.85521133 rad
    Longitude Natural Origin        2o00'00""W  = -0.03490659 rad
    Scale factor ko                     0.9996013                                                                                              False Eastings FE                 400000.00 m
    False Northings FN              -100000.00 m

    Forward calculation for:
    Latitude       50o30'00.00""N  = 0.88139127 rad
    Longitude    00o30'00.00""E  = 0.00872665 rad
    A  = 0.02775415       C = 0.00271699
    T =  1.47160434       M = 5596050.46
    M0 = 5429228.60     nu  = 6390266.03

    Then Easting E =        577274.99 m
              Northing N =       69740.50 m

    Reverse calculations for same easting and northing first gives :
    e1 =    0.00167322      mu1 = 0.87939562
    M1 = 5599036.80        nu1 = 6390275.88
    phi1  = 0.88185987      D = 0.02775243
    rho1 =6372980.21       C1 =  0.00271391
    T1 = 1.47441726

    Then Latitude       = 50o30'00.000""N
             Longitude    = 00o30'00.000""E"