## ORDNANCE SURVEY NATIONAL GRID

**Description:
**

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 . 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 in Figure The projection cylinder cuts the ellipsoid at two quasi-sub-parallels, approximately 180 km each side of the central meridian, where the scale factor will be unity. Inside these two 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 (2◦ W) which constitutes the N-axis (Y-axis) was assigned a large easting value of E 400 000 m. The E-axis (X-axis) was assigned a value of N –100 000mat 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) . 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 γ , 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

**Scale on the projection**

The basic equation for scale factor is given in equation , where the size of the ellipsoid and the value of the scale factor on the central meridian (F0) are considered. Specific to the OSGB 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) and varies as the square of the distance from the central meridian, i.e.

where E is the difference in easting between the central meridian and the point in question:

Consider a point 180 km east or west of the central meridian where F = 1:

Thus the value of F for a point whose NG coordinates are E 638824, N 309912 is:

As already intimated in equation , 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 6 ppm 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.