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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 Snellius–Pothenot problem ： ウィキペディア英語版 Snellius–Pothenot problem The Snellius–Pothenot problem is a problem in planar surveying. Given three known points A, B and C, an observer at an unknown point P observes that the segment AC subtends an angle $\backslash alpha$ and the segment CB subtends an angle $\backslash beta$; the problem is to determine the position of the point P. (See figure; the point denoted C is between A and B as seen from P). Since it involves the observation of known points from an unknown point, the problem is an example of resection. Historically it was first studied by Snellius, who found a solution around 1615. ==Formulating the equations== First equation Denoting the (unknown) angles ''CAP'' as ''x'' and ''CBP'' as ''y'' we get: :$x+y\; =\; 2\; \backslash pi\; -\; \backslash alpha\; -\; \backslash beta\; -\; C$ by using the sum of the angles formula for the quadrilateral ''PACB''. The variable ''C'' represents the (known) internal angle in this quadrilateral at point ''C''. (Note that in the case where the points ''C'' and ''P'' are on the same side of the line ''AB'', the angle C will be greater than $\backslash pi$). Second equation Applying the law of sines in triangles PAC and PBC we can express PC in two different ways: :$\backslash frac\; =\; \backslash rm\; =\; \backslash frac.$ A useful trick at this point is to define an auxiliary angle $\backslash phi$ such that :$\backslash tan\; \backslash phi\; =\; \backslash frac.$ (A minor note: we should be concerned about division by zero, but consider that the problem is symmetric, so if one of the two given angles is zero we can, if needed, rename that angle alpha and call the other (non-zero) angle beta, reversing the roles of A and B as well. This will suffice to guarantee that the ratio above is well defined. An alternative approach to the zero angle problem is given in the algorithm below.) With this substitution the equation becomes :$\backslash frac=\backslash tan\; \backslash phi.$ We can use two known trigonometric identities, namely :$\backslash tan\; \backslash left(\backslash frac-\backslash phi\backslash right)\; =\; \backslash frac$ and :$\backslash frac=\backslash frac$ to put this in the form of the second equation we need: :$\backslash tan\; \backslash frac(x-y)\; =\; \backslash tan\; \backslash frac(\backslash alpha+\backslash beta+C)\; \backslash tan\; \backslash left(\backslash frac-\backslash phi\backslash right).$ We now need to solve these two equations in two unknowns. Once ''x'' and ''y'' are known the various triangles can be solved straightforwardly to determine the position of P.〔Bowser: A treatise〕 The detailed procedure is shown below. 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「Snellius–Pothenot 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.