h = (z + γ) / sin γ = (z + γ′) / sin γ′.
If this is applied to the case of a map of South Africa between the limits 15° S. and 35° S. (see fig. 16) it will be found that the parallel of maximum error is 25° 20′; the errorless parallels, to the nearest degree, are those of 18° and 32°. The greatest scale error in this case is about 0.7%.
In the above account the earth has been treated as a sphere. Of course its real shape is approximately a spheroid of revolution, and the values of the axes most commonly employed are those of Clarke or of Bessel. For the spheroid, formulae arrived at by the same principles but more cumbrous in shape must be used. But it will usually be sufficient for the selection of the errorless parallels to use the simple spherical formulae given above; then, having made the selection of these parallels, the true spheroidal lengths along the meridians between them can be taken out of the ordinary tables (such as those published by the Ordnance Survey or by the U.S. Coast and Geodetic Survey). Thus, if a1, a2, are the lengths of 1° of the errorless parallels (taken from the tables), d the true rectified length of the meridian arc between them (taken from the tables),
h = { (a2 − a1) / d} 180 / π,
and the radius on paper of parallel, a1 is a1d/(a2 − a1), and the radius of any other parallel = radius of a1 ± the true meridian distance between the parallels.
This class of projection was used for the 1/1,000,000 Ordnance map of the British Isles. The three maximum scale errors in this case work out to 0.23%, the range of the projection being from 50° N. to 61° N., and the errorless parallels are 59° 31′ and 51°44′.
Where no great refinement is required it will be sufficient to take the errorless parallels as those distant from the extreme parallels about one-sixth of the total range in latitude. Thus suppose it is required to plot a projection for India between latitudes 8° and 40° N. By this rough rule the errorless parallels should be distant from the extreme parallels about 32°/6, i.e. 5° 20′; they should therefore, to the nearest degree, be 13° and 35° N. The maximum scale errors will be about 2%.
The scale errors vary approximately as the square of the range of latitude; a rough rule is, largest scale error = L2/50,000, where L is the range in the latitude in degrees. Thus a country with a range of 7° in latitude (nearly 500 m.) can be plotted on this projection with a maximum linear scale error (along a parallel) of about 0.1%;[29] there is no error along any meridian. It is immaterial with this projection (or with any conical projection) what the extent in longitude is. It is clear that this class of projection is accurate, simple and useful.
| (From Text Book of Topographical Surveying, by permission of the Controller of H. M. Stationery Office.) |
| Fig. 16.—South Africa on a conical projection with rectified meridians and two standard parallels. Scale 800 m. to 1 in. |
In the projections designated by (c) and (d) above, absolute errors of length are considered in the place of errors of scale, i.e. between any two meridians (c) the absolute errors of length of the extreme parallels are equated to the absolute error of length of the middle parallel. Using the same notation