For lines of any length the formulae of F.W. Bessel (Astr. Nach., 1823, iv. 241) are suitable.
If the two points A and B be defined by their geographical co-ordinates, we can accurately calculate the corresponding astronomical azimuths, i.e. those of the vertical section, and then proceed, in the case of not too great distances, to determine the length and the azimuth of the shortest lines. For any distances recourse must again be made to Bessel’s formula.[4]
Let α, α′ be the mutual azimuths of two points A, B on a spheroid, k the chord line joining them, μ, μ′ the angles made by the chord with the normals at A and B, φ, φ′, ω their latitudes and difference of longitude, and (x² + y²)/a² + z² b² = 1 the equation of the surface; then if the plane xz passes through A the co-ordinates of A and B will be
| x = (a/Δ) cos φ, | x′ = (a/Δ’) cos φ′ cos ω, |
| y = 0 | y′ = (a/Δ’) cos φ′ sin ω, |
| z = (a/Δ) (1 − e²) sin φ, | z′ = (a/Δ′) (1 − e²) sin φ′, |
where Δ = (1 − e² sin² φ)1/2, Δ′ = (1 − e² sin² φ′)1/2, and e is the eccentricity. Let f, g, h be the direction cosines of the normal to that plane which contains the normal at A and the point B, and whose inclinations to the meridian plane of A is = α; let also l, m, n and l’, m’, n’ be the direction cosines of the normal at A, and of the tangent to the surface at A which lies in the plane passing through B, then since the first line is perpendicular to each of the other two and to the chord k, whose direction cosines are proportional to x′ − x, y′ − y, z′ − z, we have these three equations
| f (x′ − x) + gy′ + h (z′ − z) = 0 |
| fl + gm + hn = 0 |
| fl′ + gm′ + hn′ = 0. |
Eliminate f, g, h from these equations, and substitute
| l = cos φ | l′ = − sin φ cos α |
| m = 0 | m′ = sin α |
| n = sin φ | n′ = cos φ cos α, |
and we get
(x′ − x) sin φ + y′ cot α − (z′ − z) cos φ = 0.