Geodesics on an ellipsoid

The study of geodesics on an ellipsoid arose in connection with geodesyspecifically with the solution of triangulation networks. Thefigure of the Earth is well approximated by anoblate ellipsoid, a slightly flattened sphere. A geodesicis the shortest path between two points on a curved surface, analogousto a straight line on a plane surface. The solution of a triangulationnetwork on an ellipsoid is therefore a set of exercises in spheroidaltrigonometry (Euler 1755).Nous désignerons cette ligne sous le nom de ligne géodésique [Wewill call this line the geodesic line].A line traced in the manner we have now been describing, or deduced fromtrigonometrical measures, by the means we have indicated, is calleda geodetic or geodesic line: it has the property of beingthe shortest which can be drawn between its two extremities on thesurface of the Earth; and it is therefore the proper itinerarymeasure of the distance between those two points. The day before yesterday, I reduced to quadrature the problem of geodesic lines on an ellipsoid with three unequal axes. They are the simplest formulas in the world, Abelian integrals, which become the well known elliptic integrals if 2 axes are set equal. The study of geodesics on an ellipsoid arose in connection with geodesyspecifically with the solution of triangulation networks. Thefigure of the Earth is well approximated by anoblate ellipsoid, a slightly flattened sphere. A geodesicis the shortest path between two points on a curved surface, analogousto a straight line on a plane surface. The solution of a triangulationnetwork on an ellipsoid is therefore a set of exercises in spheroidaltrigonometry (Euler 1755). If the Earth is treated as a sphere, the geodesics aregreat circles (all of which are closed) and the problems reduce toones in spherical trigonometry. However, Newton (1687)showed that the effect of the rotation of the Earth results in itsresembling a slightly oblate ellipsoid and, in this case, theequator and the meridians are the only simpleclosed geodesics. Furthermore, the shortest path between two points onthe equator does not necessarily run along the equator. Finally, if theellipsoid is further perturbed to become a triaxial ellipsoid (withthree distinct semi-axes), only three geodesics are closed. There are several ways of defining geodesics(Hilbert & Cohn-Vossen 1952, pp. 220–221). A simple definitionis as the shortest path between two points on a surface. However, it isfrequently more useful to define them as paths with zerogeodesic curvature—i.e., the analogue of straight lines on acurved surface. This definition encompasses geodesics traveling so faracross the ellipsoid's surface (somewhat more than half thecircumference) that other distinct routes require less distance.Locally, these geodesics are still identical to the shortest distancebetween two points. By the end of the 18th century, an ellipsoid of revolution (the termspheroid is also used) was a well-accepted approximation to thefigure of the Earth. The adjustment of triangulation networksentailed reducing all the measurements to a reference ellipsoid andsolving the resulting two-dimensional problem as an exercise inspheroidal trigonometry (Bomford 1952, Chap. 3)(Leick et al. 2015, §4.5). It is possible to reduce the various geodesic problems into one of twotypes. Consider two points: A at latitudeφ1 and longitude λ1 andB at latitude φ2 and longitudeλ2 (see Fig. 1). The connecting geodesic(from A to B) is AB, of lengths12, which has azimuths α1 andα2 at the two endpoints. The two geodesic problems usuallyconsidered are: As can be seen from Fig. 1, these problems involve solving the triangleNAB given one angle, α1 for the directproblem and λ12 = λ2 − λ1 for theinverse problem, and its two adjacent sides.For a sphere the solutions to these problems are simple exercises inspherical trigonometry, whose solution is given byformulasfor solving a spherical triangle.(See the article on great-circle navigation.) For an ellipsoid of revolution, the characteristic constant defining thegeodesic was found by Clairaut (1735). Asystematic solution for the paths of geodesics was given byLegendre (1806) andOriani (1806) (and subsequent papers in1808 and1810).The full solution for the direct problem (complete with computationaltables and a worked out example) is given by Bessel (1825). During the 18th century geodesics were typically referred to as 'shortestlines'.The term 'geodesic line' was coined by Laplace (1799b): This terminology was introduced into English either as 'geodesic line'or as 'geodetic line', for example (Hutton 1811),