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3.4.4.4 Transverse Mercator

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.

Table 3-3 below 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:

Table 3-3: Transverse Mercator

 The most familiar and commonly used Transverse Mercator in the oil industry 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:

Northing:

where

, with l and l₀ in radians

with j in radians and M₀ for j₀, 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:

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:

Forward calculation for:

gives the following easting and northing:

Reverse the calculations. Use the same parameters and use the E and N values to calculate the latitude and longitude:

Then

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