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# 7: Transverse Mercator map projections

When features on the curved surface of the Earth are represented on a plane surface, distortions of distances, angles or both are inevitable. Originally, the plane surface was a map sheet; now it is often the plane coordinate system of GIS software. A map projection is any function which converts ellipsoidal latitude and longitude coordinates to plane easting and northing coordinates. Ordnance Survey maps use a type of projection known as the Transverse Mercator (TM). The same type of projection is used in a worldwide mapping standard known as Universal Transverse Mercator (UTM). The parameters of the National Grid TM and UTM projections are given in annexe A.

The TM projection can be thought of as a sheet of paper carrying the mapping grid (of eastings and northings), which is curved so as to touch the ellipsoid along a certain line. This line of contact is chosen to be a north-south central meridian. Points on the ellipsoid are projected onto the curved sheet, giving easting and northing coordinates for each point. The effect is to distort the distance between projected points, except on the central meridian where the ellipsoid touches the mapping grid. This scale distortion effect increases east and west of the central meridian. The scale distortion can be measured by a local scale factor which is 1 on the central meridian and greater than 1 everywhere else.

To reduce the worst scale distortion effect in the extreme eastern and western regions of the mapping area, a scale reduction factor is introduced over the whole mapping area. This makes projected distances on the central meridian slightly too small, but lessens the scale distortion for points far to the east or west of the central meridian. With the overall scale reduction applied, there are now two lines (either side of the central meridian) on which the local scale factor is 1. Inside these lines, the local scale factor is less than 1 (with a minimum on the central meridian), and outside these lines it is more than 1. On the central meridian, the local scale factor is now equal to the scale reduction factor that was introduced.

The TM projection used for all Ordnance Survey maps has a central meridian at longitude 2° West and a central meridian scale factor of approximately 0.9996 (see Annex A for exact value). The two lines of true scale are about 180 km to the east and west of the central meridian. The stated scale of an Ordnance Survey map is only exactly true on these lines of true scale, but the scale error elsewhere is quite small. For instance, the true scale of Ordnance Survey 1:50 000 scale map sheets is actually between 1:49 980 and 1:50 025 depending on easting.

The UTM projections are a way of mapping the whole world in a systematic way by dividing the Earth by longitude into 60 zones, each 6 degrees of longitude wide. The 60 UTM zones each have a different central meridian. The zones relevant to British mapping are 29 (central meridian 9° W), 30 (central meridian 3° W) and 31 (central meridian 3° E). The scale on the central meridian is 0.9996 for all UTM zones. The International 1924 ellipsoid is usually used in the UTM projection in Europe, but other ellipsoids can also be used. When working with UTM coordinates, check which ellipsoid is being used.

Annexe C gives formulae for conversion between latitude and longitude and TM eastings and northings. Annexe A gives the TM parameters for the National Grid and the British UTM zones.

7.1 The National Grid reference convention

The map projection used on Ordnance Survey Great Britain maps is known as the National Grid. The Transverse Mercator eastings and northings axes are given a 'false origin' just south-west of the Scilly Isles to ensure that all coordinates in Britain are positive. The false origin is 400 km west and 100 km north of the true origin on the central meridian at 49° N 2° W.

To reduce the number of figures needed to give a National 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 between 0 and 100 000 m and a northing between 0 and 100 000 m.

Figure 8: The National Grid, showing the true origin (t.o.) and false origin (f.o.)

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