It is rarely that the terms of the fourth order are required. As a rule spheroidal triangles are calculated as spherical (after Legendre), i.e. like plane triangles with a decrease of each angle of about ε/3; ε must, however, be calculated for each triangle separately with its mean measure of curvature k.
The geodetic line being the shortest that can be drawn on any surface between two given points, we may be conducted to its most important characteristics by the following considerations: let p, q be adjacent points on a curved surface; through s the middle point of the chord pq imagine a plane drawn perpendicular to pq, and let S be any point in the intersection of this plane with the surface; then pS + Sq is evidently least when sS is a minimum, which is when sS is a normal to the surface; hence it follows that of all plane curves on the surface joining p, q, when those points are indefinitely near to one another, that is the shortest which is made by the normal plane. That is to say, the osculating plane at any point of a geodetic line contains the normal to the surface at that point. Imagine now three points in space, A, B, C, such that AB = BC = c; let the direction cosines of AB be l, m, n, those of BC l’, m′, n′, then x, y, z being the co-ordinates of B, those of A and C will be respectively—
| x − cl : y − cm : z − cn x + cl′ : y + cm′ : z + cn′. |
Hence the co-ordinates of the middle point M of AC are x + ½c(l′ − l), y + ½c(m′ − m), z + ½c(n′ − n), and the direction cosines of BM are therefore proportional to l′ − l: m′ − m: n′ − n. If the angle made by BC with AB be indefinitely small, the direction cosines of BM are as δl : δm : δn. Now if AB, BC be two contiguous elements of a geodetic, then BM must be a normal to the surface, and since δl, δm, δn are in this case represented by δ(dx/ds), δ(dy/ds), δ(dz/ds), and if the equation of the surface be u = 0, we have
| d²x | / | du | = | d²y | / | du | = | d²z | / | du | , |
| ds² | dx | ds² | dy | ds² | dz |
which, however, are equivalent to only one equation. In the case of the spheroid this equation becomes
| y | d²x | − | d²y | = 0, |
| ds² | ds² |
which integrated gives ydx − xdy = Cds. This again may be put in the form r sin a = C, where a is the azimuth of the geodetic at any point—the angle between its direction and that of the meridian—and r the distance of the point from the axis of revolution.
From this it may be shown that the azimuth at A of the geodetic joining AB is not the same as the astronomical azimuth at A of B or that determined by the vertical plane AαB. Generally speaking, the geodetic lies between the two plane section curves joining A and B which are formed by the two vertical planes, supposing these points not far apart. If, however, A and B are nearly in the same latitude, the geodetic may cross (between A and B) that plane curve which lies nearest the adjacent pole of the spheroid. The condition of crossing is this. Suppose that for a moment we drop the consideration of the earth’s non-sphericity, and draw a perpendicular from the pole C on AB, meeting it in S between A and B. Then A being that point which is nearest the pole, the geodetic will cross the plane curve if AS be between ¼AB and 3⁄8AB. If AS lie between this last value and ½AB, the geodetic will lie wholly to the north of both plane curves, that is, supposing both points to be in the northern hemisphere.
The difference of the azimuths of the vertical section AB and of the geodetic AB, i.e. the astronomical and geodetic azimuths, is very small for all observable distances, being approximately:—