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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hansen's problem ： ウィキペディア英語版 Hansen's problem Hansen's problem is a problem in planar surveying, named after the astronomer Peter Andreas Hansen (1795–1874), who worked on the geodetic survey of Denmark. There are two known points ''A'' and ''B'', and two unknown points ''P''1 and ''P''2. From ''P''1 and ''P''2 an observer measures the angles made by the lines of sight to each of the other three points. The problem is to find the positions of ''P''1 and ''P''2. See figure; the angles measured are (''α''1, ''β''1, ''α''2, ''β''2). Since it involves observations of angles made at unknown points, the problem is an example of resection (as opposed to intersection). == Solution method overview== Define the following angles: ''γ'' = ''P''1''AP''2, ''δ'' = ''P''1''BP''2, ''φ'' = ''P''2''AB'', ''ψ'' = ''P''1''BA''. As a first step we will solve for ''φ'' and ''ψ''. The sum of these two unknown angles is equal to the sum of ''β''1 and ''β''2, yielding the following equation: :$\backslash phi+\backslash psi=\backslash beta\_1+\backslash beta\_2$ A second equation can be found more laboriously, as follows. The law of sines yields :$\backslash frac=\backslash frac$ and :$\backslash frac=\backslash frac$ combining these together we get :$\backslash frac=\backslash frac$ An entirely analogous reasoning on the other side yields :$\backslash frac=\backslash frac$ Setting these two equal gives :$\backslash frac=\backslash frac\; =\; k$ Using a known trigonometric identity this ratio of sines can be expressed as the tangent of an angle difference: :$\backslash tan\; \backslash frac=\backslash frac\backslash tan\backslash frac$ This is the second equation we need. Once we solve the two equations for the two unknowns $\backslash phi$ and $\backslash psi$, we can use either of the two expressions above for $\backslash frac$ to find ''P''1''P''2 since ''AB'' is known. We can then find all the other segments using the law of sines.〔Udo Hebisch: Ebene und Sphaerische Trigonometrie, Kapitel 1, Beispiel 4 (2005, 2006)()〕 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「Hansen's problem」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.