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The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an ''oblate ellipsoid'', a slightly flattened sphere. A ''geodesic'' is the shortest path between two points on a curved surface, i.e., the analogue of a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry . If the Earth is treated as a sphere, the geodesics are great circles (all of which are closed) and the problems reduce to ones in spherical trigonometry. However, showed that the effect of the rotation of the Earth results in its resembling a slightly oblate ellipsoid and, in this case, the equator and the meridians are the only closed geodesics. Furthermore, the shortest path between two points on the equator does not necessarily run along the equator. Finally, if the ellipsoid is further perturbed to become a triaxial ellipsoid (with three distinct semi-axes), then only three geodesics are closed and one of these is unstable. The problems in geodesy are usually reduced to two main cases: the ''direct problem'', given a starting point and an initial heading, find the position after traveling a certain distance along the geodesic; and the ''inverse problem'', given two points on the ellipsoid find the connecting geodesic and hence the shortest distance between them. Because the flattening of the Earth is small, the geodesic distance between two points on the Earth is well approximated by the great-circle distance using the mean Earth radius—the relative error is less than 1%. However, the course of the geodesic can differ dramatically from that of the great circle. As an extreme example, consider two points on the equator with a longitude difference of 179°59′; while the connecting great circle follows the equator, the shortest geodesics pass within 180 km of either pole (the flattening makes two symmetric paths passing close to the poles shorter than 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 signals which are confined (approximately) to the surface of the Earth, for example, sound waves in the ocean and the radio signals from lightning . Geodesics are used to define some maritime boundaries, which in turn determine the allocation of valuable resources as such oil and mineral rights. Ellipsoidal geodesics also arise in other applications; for example, the propagation of radio waves along the fuselage of an aircraft, which can be roughly modeled as a prolate (elongated) ellipsoid . Geodesics are an important intrinsic characteristic of curved surfaces. The sequence of progressively more complex surfaces, the sphere, an ellipsoid of revolution, and a triaxial ellipsoid, provide a useful family of surfaces for investigating the general theory of surfaces. Indeed, Gauss's work on the survey of Hanover, which involved geodesics on an oblate ellipsoid, was a key motivation for his study of surfaces . Similarly, the existence of three closed geodesics on a triaxial ellipsoid turns out to be a general property of closed, simply connected surfaces; this theorem of the three geodesics was conjectured by and proved by . == Geodesics on an ellipsoid of revolution == There are several ways of defining geodesics . A simple definition is as the shortest path between two points on a surface. However, it is frequently more useful to define them as paths with zero geodesic curvature—i.e., the analogue of straight lines on a curved surface. This definition encompasses geodesics traveling so far across the ellipsoid's surface (somewhat less than half the circumference) that other distinct routes require less distance. Locally, these geodesics are still identical to the shortest distance between two points. By the end of the 18th century, an ellipsoid of revolution (the term spheroid is also used) was a well-accepted approximation to the figure of the Earth. The adjustment of triangulation networks entailed reducing all the measurements to a reference ellipsoid and solving the resulting two-dimensional problem as an exercise in spheroidal trigonometry . It is possible to reduce the various geodesic problems into one of two types. Consider two points: ''A'' at latitude φ1 and longitude λ1 and ''B'' at latitude φ2 and longitude λ2 (see Fig. 1). The connecting geodesic (from ''A'' to ''B'') is ''AB'', of length ''s''12, which has azimuths α1 and α2 at the two endpoints. The two geodesic problems usually considered are: # the ''direct geodesic problem'' or ''first geodesic problem'', given ''A'', α1, and ''s''12, determine ''B'' and α2; # the ''inverse geodesic problem'' or ''second geodesic problem'', given ''A'' and ''B'', determine ''s''12, α1, and α2. As can be seen from Fig. 1, these problems involve solving the triangle ''NAB'' given one angle, α1 for the direct problem and λ12 = λ2 − λ1 for the inverse problem, and its two adjacent sides. In the course of the 18th century these problems were elevated (especially in literature in the German language) to the ''principal geodesic problems'' . For a sphere the solutions to these problems are simple exercises in spherical trigonometry, whose solution is given by formulas for solving a spherical triangle. (See the article on great-circle navigation.) For an ellipsoid of revolution, the characteristic constant defining the geodesic was found by . A systematic solution for the paths of geodesics was given by and (and subsequent papers in and ). The full solution for the direct problem (complete with computational tables and a worked out example) is given by . Much of the early work on these problems was carried out by mathematicians—for example, Legendre, Bessel, and Gauss—who were also heavily involved in the practical aspects of surveying. Beginning in about 1830, the disciplines diverged: those with an interest in geodesy concentrated on the practical aspects such as approximations suitable for field work, while mathematicians pursued the solution of geodesics on a triaxial ellipsoid, the analysis of the stability of closed geodesics, etc. During the 18th century geodesics were typically referred to as "shortest lines". The term "geodesic line" was coined by :
This terminology was introduced into English either as "geodesic line" or as "geodetic line", for example ,
In its adoption by other fields "geodesic line", frequently shortened, to "geodesic", was preferred. This section treats the problem on an ellipsoid of revolution (both oblate and prolate). The problem on a triaxial ellipsoid is covered in the next section. When determining distances on the earth, various approximate methods are frequently used; some of these are described in the article on geographical distance. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「geodesics on an ellipsoid」の詳細全文を読む スポンサード リンク
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