翻訳と辞書 Words near each other ・ Legendre chi function ・ Legendre form ・ Legendre function ・ Legendre polynomials ・ Legendre pseudospectral method ・ Legendre rational functions ・ Legendre relation ・ Legendre sieve ・ Legendre symbol ・ Legendre transformation ・ Legendre wavelet ・ Legendre's conjecture ・ Legendre's constant ・ Legendre's equation ・ Legendre's formula ・ Legendre's theorem on spherical triangles ・ Legendre's three-square theorem ・ Legendre–Clebsch condition ・ Legendrian knot ・ Legends (Above the Law album) ・ Legends (Beverley Craven album) ・ Legends (Bob Catley album) ・ Legends (book) ・ Legends (comics) ・ Legends (Dvořák) ・ Legends (Five Star album) ・ Legends (PBM) ・ Legends (Transformers) ・ Legends (TV series) ・ Legends about Theoderic the Great Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Legendre's theorem on spherical triangles ： ウィキペディア英語版 Legendre's theorem on spherical triangles In geometry, Legendre's theorem on spherical triangles, named after Adrien-Marie Legendre, is stated as follows: : Let ABC be a spherical triangle on the ''unit'' sphere with ''small'' sides ''a'', ''b'', ''c''. Let A'B'C be the planar triangle with the same sides. Then the angles of the spherical triangle exceed the corresponding angles of the planar triangle by approximately one third of the spherical excess (the spherical excess is the amount by which the sum of the three angles exceeds ). The theorem was very important in simplifying the heavy numerical work in calculating the results of traditional (pre-GPS and pre-computer) geodetic surveys from about 1800 until the middle of the twentieth century. The theorem was stated by who provided a proof in a supplement to the report of the measurement of the French meridional arc used in the definition of the metre . Legendre does not claim that he was the originator of the theorem despite the attribution to him. maintains that the method was in common use by surveyors at the time and may have been used as early as 1740 by La Condamine for the calculation of the Peruvian meridional arc. Girard's theorem states that the spherical excess of a triangle, ''E'', is equal to its area, Δ, and therefore Legendre's theorem may be written as ::$$ \begin A-A'\;\approx\; B-B'\;\approx\; C-C'\;\approx\;\frac13 E\;=\; \frac13\Delta,\qquad a,\;b,\;c\,\ll\, 1. \end The excess, or area, of small triangles is very small. For example consider an equilateral spherical triangle with sides of 60 km on a spherical Earth of radius 6371 km; the side corresponds to an angular distance of 60/6371=.0094, or approximately 10−2 radians (subtending an angle of 0.57° at the centre). The area of such a small triangle is well approximated by that of a planar equilateral triangle with the same sides: ''a''2sin(/3) = 0.0000433 radians corresponding to 8.9″. When the sides of the triangles exceed 180 km, for which the excess is about 80″, the relations between the areas and the differences of the angles must be corrected by terms of fourth order in the sides, amounting to no more than 0.01″: ::$\backslash begin$ \Delta &=\Delta'\left( 1+\frac \right),\\ A&=A'+\frac +\frac \left( -2a^2+b^2+c^2 \right),\\ B&=B'+\frac +\frac \left( \right),\\ C&=C'+\frac +\frac \left( \right). \end (Δ′ is the area of the planar triangle.) This result was proved by —an extended proof may be found in (Appendix D13). Other results are surveyed by . The theorem may be extended to the ellipsoid if ''a'', ''b'', ''c'' are calculated by dividing the true lengths by the square root of the product of the principal radii of curvature (see Chapter 5) at the median latitude of the vertices (in place of a spherical radius). provided more exact formulae. ==References== * * * * * * * * 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「Legendre's theorem on spherical triangles」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.