The Universal Transverse Mercator Projection 

The Universal Transverse Mercator projection ( UTM ) is a worldwide system of transverse Mercator projection.

It comprises 60 zones , each 6 wide in longitude , with central meridians at 3 , 9 , etc The zones are numbered from 1 to 60 , starting with 180 to 174 W as zone 1 and proceeding eastwards to zone 60. 

Therefore the central meridian ( CM ) of zone n is given by CM = 6n - 183 In latitude ,the UTM system extends from 84 N to 80 S , with the polar stereographic projection. 

The scale factor at each central meridian is 0.9996 to counteract the enlargement ratio at the edges of the strips The false origin of northings is zero at the equator for the northern hemisphere and 10 m at the equater for the southern hemisphere The false origin for eastings is 5 × 10 m west of the zone central meridian. 

Ordnance Survey National Grid 

The Ordnance Survey (OS ) is the national mapping agency for Great Britain ; its maps are based on a transverse Mercator projection of Airy’s ellipsoid called the OSGB (36) datum the current realization of OSGB (36) is the OS’s Terrestrial Network 2002 ( OSTN02 ) datum which is a rubber sheet fit of European Terrestrial Reference System 1989 ( ETRS89 ) coordinates as derived from GPS to the original OSGB (36 ) For most practical purposes there should be no significant difference between OSGB ( 36 ) and OSTN02 the central meridian selected is 2 W with the point of origin called the false origin at 49 N on this meridian the scale factor varies as the square of the distance from the central meridian and therefore in order to reduce scale error at the extreme east and west edges of the country the scale factor on the central meridian was reduced to 0.999 601 27 one can think of this as reducing the radius of the enclosing cylinder as shown.

The projection cylinder cuts the ellipsoid at two quasi sub parallels the scale is too small by less than 0.04% and outside of them too large by up to 0.05% on the west coast of mainland Scotland 

The central meridian (2W) which constitutes the N axes ( Y axes ) was assigned a large easting value of e 400 000 m the E axes (X axes ) was assigned a value of N 100 000 m at the 49 N parallel of latitude on the CM thus a rectangular grid is superimposed on the developed cylinder and is called the OS National Grid (NG) (Figure 8.19 ) the assigned values result in a false origin and positive values only throughout what is now a plane rectangular coordinate system such a grid thereby establishes the direction of grid north which differs from geodetic north by y , a variable amount called the grid convergence on the central meridian grid north and geodetic north are the same direction. 

Scale factors 

The concept of scale factors has been fully dealt with and it only remains to deal with their application It should be clearly understood that scale factors transform distance on the ellipsoid to distance on the. 

plane of projection from 

 it can be seen that a horizontal distance at ground level AB must first be reduced to its equivalent at MSL ( geoid ) A B using the altitude correction thence to the ellipsoid A B using the geoid ellipsoid value ( N ) and then multiplied by the scale factor to produce the projection distance A2 B2 Whilst this is theoretically the correct approach lack of knowledge of N may result in this step being ignored in great Britain the maximum value is 4.5 m resulting in a scale error of only 0.7 ppm if ignored thus the practical approach is to reduce to MSL and then to the projection plane i e from D to S to G.

The basic equation for scale factor

The basic equation for scale factor is given in equation (8,46) where the size of the ellipsoid and the value of the scale factor on the central neridian ( F0 ) are considered specific to the OSGB ( 36 ) system the following formula may be developed which is sufficiently accurate for most purposes 

Scale difference ( SD ) is the difference between the scale factor at any point ( F )  and that at the central meridian ( F0 ) are varies as the square of the distance from the central meridian i e.

National reference system of Great Britain showing 100 km squares the figures used to designate them in the former system and the letters which have replaced the figures country Ordnance Survey Crown Copyright Reserved.

As already intimated in equation ( 8.46 ) the treatment for highly accurate work is to compute F for each end of the line and in the middle and then obtain the mean value from Simpson’s rule however for most practical purposes on short lines it is sufficient to compute F at the mid point of a line in OSGB ( 36 ) the scale factor varies at the most by only 6nppm per km and hence a single value for F at the centre of a small site can be regarded as constant throughout the area on long motorway or route projects however one would need to use different scale factors for different sections.

The following examples will server to illustrate the classical application of scale factors