transverse

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





where j1 may be found as for the Cassini projection from:





and where













, with n1 = n for j1

For areas south of the equator the value of latitude j 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 10,000,000 m. 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 500,000 m 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 j results from a negative j1 which in turn results from a negative M1.

For the mapping of southern Africa a south oriented Transverse Mercator projection is used. Here the coordinate axes are called Westings and Southings and increment to the West and South from the origin respectively. See Figure 3-6 below for a diagrammatic illustration. The above formulas need to be modified to cope with this arrangement with

Westing:



Southing:





In these formulas the terms FE and FN have been retained for consistency of the terminology. For the reverse formulas, those for the standard Transverse Mercator above apply, with the exception that:



and, with n1 = n for j1

Example:
In addition to the example below, a form, UTM Calculation Sheet, has been developed that allows you to perform a UTM transformation on a limited number of datums.

Example:
For Projected Coordinate System OSGB 1936 / British National Grid

Parameters:

Ellipsoid Airy 1830 a = 6377563.396 m  
 
  
 1/f = 299.32496
 
 
  
 e'2 = 0.00671534
 
 
  
 e2 = 0.00667054
 
Latitude of Natural Origin
 j0
 49°00' 00"N =
 0.85521133 rad
 
Longitude of Natural Origin
 lF
 2°00' 00"W =
 -0.03490659 rad
 
Scale factor
 k0
 0.9996013
  
 
False Eastings
 FE
 400000.00 m
  
 
False Northings
 FN
 -100000.00 m
  
 

Forward calculation for:

Latitude
 j
 50°30' 00.00" N =
 0.88139127 rad
 
Longitude
 l
 00°30' 00.00" E =
 0.00872665 rad
 

 

A =
 0.02775415
 C =
 0.00271699
 
T =
 1.47160434
 M =
 5596050.46
 
n =
 6390266.03
 M0 =
 5429228.60
 

gives the following easting and northing: