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, i.e., the analogueof 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 onlyclosed 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), then only three geodesics are closed and oneof these is unstable.The problems in geodesy are usually reduced to two main cases: thedirect problem, given a starting point and an initial heading, findthe position after traveling a certain distance along the geodesic; andthe inverse problem, given two points on the ellipsoid find theconnecting geodesic and hence the shortest distance between them.Because the flattening of the Earth is small, the geodesic distancebetween two points on the Earth is well approximated by the great-circledistance using themean Earth radius—the relative error isless than 1%. However, the course of the geodesic can differdramatically from that of the great circle. As an extreme example,consider two points on the equator with a longitude difference of179°59′; while the connecting great circle follows theequator, the shortest geodesics pass within180 km of either pole (theflattening makes two symmetric paths passing close to the poles shorterthan the route along the equator).Aside from their use in geodesy and related fields such as navigation,terrestrial geodesics arise in the study of the propagation of signalswhich are confined (approximately) to the surface of the Earth, forexample, sound waves in the ocean (Munk & Forbes 1989) and theradio signals from lightning (Casper & Bent 1991). Geodesics areused to define some maritime boundaries, which in turn determine theallocation of valuable resources as suchoil and mineral rights. Ellipsoidal geodesics alsoarise in other applications; for example, the propagation of radio wavesalong the fuselage of an aircraft, which can be roughly modeled as aprolate (elongated) ellipsoid(Kim & Burnside 1986).Geodesics are an important intrinsic characteristic of curved surfaces.The sequence of progressively more complex surfaces, the sphere, anellipsoid of revolution, and a triaxial ellipsoid, provide a usefulfamily of surfaces for investigating the general theory of surfaces.Indeed, Gauss's work on thesurvey of Hanover, which involvedgeodesics on an oblate ellipsoid, was a key motivation for hisstudy of surfaces(Gauss 1828). Similarly, the existence of three closed geodesicson a triaxial ellipsoid turns out to be a general property ofclosed, simply connected surfaces; this wasconjectured by Poincaré (1905) and proved byLyusternik & Schnirelmann (1929)(Klingenberg 1982, §3.7).