When searching by computer for long-distance alinements (e.g. leys, dragon lines, fire leynes) it is necessary to allow for the fact that a geodesic does not in general project onto a straight line in the National Grid. We could apply the required correction to each line as it is found. A simpler and more efficient method is to adjust the site coordinates before beginning the search, as explained below.
We know that for ley-hunting the Earth’s polar flattening can be ignored. More precisely: the Ordnance Survey give us a Transverse Mercator projection of the spheroid onto a plane; if we project the plane by Transverse Mercator onto a perfect sphere of suitable radius, then the image of Britain on the sphere is negligibly different from Britain on the spheroid. This fact was used to derive the formulae in the 1977 notes. It is found empirically that a good value for the sphere’s radius R is such that
|
This quantity often appears in the formulae.
The Transverse Mercator projection of the sphere does not in general send geodesics into straight lines. But there is a projection of the sphere that does so, namely the Central projection defined as follows: let O be the centre of the sphere, T a plane not through O; then the image of a point P on the sphere is the intersection of OP and T.
Before beginning the computer search, we can convert the National Grid coordinates of each site into coordinates in such a plane T; then geodesics will become (for practical purposes) straight lines. It is natural to choose T as tangent to the sphere at a point in the centre of Britain. We shall use here a point on the central meridian, having NG coordinates E=400 km, N=500 km (the point NZ 00).
Let (E,N) be NG coordinates and let (λ,φ) be longitude and latitude on the sphere, with λ measured from the central meridian of the projection (2° W). The Transverse Mercator projection of the sphere is given by
|
(1) |
where E0=400 km and N1 is some constant. Let the plane T touch the sphere on the central meridian; let the point of contact have NG coordinates (E0,N0) and geographical coordinates (0,φ0). In T take the origin at that point, X eastwards, Y northwards. Then the Central projection of the sphere is given by
|
(2) |
where W=sinφ sinφ0+cosλ cosφ cosφ0.
We can now express (X,Y) in terms of (E,N). The N1 drops out and we get simply
|
(3) |
As in the 1977 notes, put D=E−E0; also put M=N−N0 and k=1/6R2. For ley-hunting we can ignore terms in 1/R4, so that (3) becomes
|
(4) |
With a unit of 100 km we shall have
|
In the tangent plane T it will be convenient to use coordinates (E′,N′) given by
|
so that the point of contact has coordinates (4, 5) in both planes.
In practice we can regard the mapping (4) as an adjustment to the NG coordinates given by
|
(5) |
where k=4.092×10−5.
The following examples confirm that the adjustment (5) does indeed transform geodesics into straight lines with sufficient accuracy for ley-hunting.
Take the geodesic to be the Greenwich meridian, with points every 1° from 50° to 60°. From the OS Projection Tables the NG coordinates of the points are found to be:
| Lat | E | N |
| 50 | 5.43315410 | 0.13081909 |
| 51 | 5.40319840 | 1.24253656 |
| 52 | 5.37281173 | 2.35442150 |
| 53 | 5.34200325 | 3.46647263 |
| 54 | 5.31078226 | 4.57868844 |
| 55 | 5.27915821 | 5.69106723 |
| 56 | 5.24714067 | 6.80360710 |
| 57 | 5.21473934 | 7.91630596 |
| 58 | 5.18196404 | 9.02916151 |
| 59 | 5.14882473 | 10.14217125 |
| 60 | 5.11533150 | 11.25533252 |
Let us apply the adjustment (5) and see how far the resulting points are from the best-fitting straight line (calculated by least squares). The result is:
| Lat | E′ | N′ | Error (m) |
| 50 | 5.43744576 | 0.12137123 | -1.029 |
| 51 | 5.40574347 | 1.23819496 | -0.243 |
| 52 | 5.37409713 | 2.35290610 | +0.141 |
| 53 | 5.34248958 | 3.46617748 | +0.316 |
| 54 | 5.31090298 | 4.57868232 | +0.409 |
| 55 | 5.27931885 | 5.69109424 | +0.486 |
| 56 | 5.24771808 | 6.80408727 | +0.548 |
| 57 | 5.21608094 | 7.91833581 | +0.539 |
| 58 | 5.18438715 | 9.03451466 | +0.345 |
| 59 | 5.15261587 | 10.15329895 | -0.200 |
| 60 | 5.12074577 | 11.27536417 | -1.311 |
The adjusted points are contained in a strip of width 1.9 metres, versus 484 metres for the unadjusted points.
One version of John Michell’s famous line is got by drawing a geodesic from St Michael’s Mount to Bury St Edmunds Cathedral. Following Bob Forrest’s notes, take three points A, B, C on this line: A and B near the points where the geodesic differs most from a straight line drawn on the map, and C near the point where the geodesic crosses that line.
(No attempt is made here to find the theoretically exact coordinates of those 3 points, but A, B, C are accurately on the geodesic as calculated from the O.S. tables.) As shown by Bob Forrest, the distance on the ground between geodesic and straight line is about 18 metres at A, 3 metres at B.
| Site | E | N |
| St M M | 1.5146 | 0.2986 |
| A | 2.70837142 | 0.94302948 |
| B | 4.62948279 | 1.97955893 |
| C | 5.29154785 | 2.33680477 |
| Bury St E | 5.8559 | 2.6414 |
As in Example 1, apply the adjustment (5) and then calculate the best-fitting line. The distance offline is cut down to about 50 mm.
| Site | E′ | N′ | Error (m) |
| St M M | 1.50722792 | 0.29009553 | -0.027 |
| A | 2.70567350 | 0.93756472 | +0.014 |
| C | 4.63019798 | 1.97730377 | +0.053 |
| B | 5.29276055 | 2.33525889 | +0.011 |
| Bury St E | 5.85742900 | 2.64032619 | -0.052 |
Michael Behrend
18 June 1986