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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 Hyperbolic coordinates ： ウィキペディア英語版 Hyperbolic coordinates In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane :$\backslash \; =\; Q\backslash \; \backslash !$. Hyperbolic coordinates take values in the hyperbolic plane defined as: :$HP\; =\; \backslash$. These coordinates in ''HP'' are useful for studying logarithmic comparisons of direct proportion in ''Q'' and measuring deviations from direct proportion. For $(x,y)$ in $Q$ take :$u\; =\; \backslash ln\; \backslash sqrt\}$ and :$v\; =\; \backslash sqrt$. Sometimes the parameter $u$ is called the hyperbolic angle and $v$ Is called the geometric mean. The inverse mapping is :$x\; =\; v\; e^u\; ,\backslash quad\; y\; =\; v\; e^$. This is a continuous mapping, but not an analytic function. ==Alternative quadrant metric== Since ''HP'' carries the metric space structure of the Poincaré half-plane model of hyperbolic geometry, the bijective correspondence $Q\; \backslash leftrightarrow\; HP$ brings this structure to ''Q''. It can be grasped using the notion of hyperbolic motions. Since geodesics in ''HP'' are semicircles with centers on the boundary, the geodesics in ''Q'' are obtained from the correspondence and turn out to be rays from the origin or petal-shaped curves leaving and re-entering the origin. And the hyperbolic motion of ''HP'' given by a left-right shift corresponds to a squeeze mapping applied to ''Q''. Since hyperbolas in ''Q'' correspond to lines parallel to the boundary of ''HP'', they are horocycles in the metric geometry of ''Q''. If one only considers the Euclidean topology of the plane and the topology inherited by ''Q'', then the lines bounding ''Q'' seem close to ''Q''. Insight from the metric space ''HP'' shows that the open set ''Q'' has only the origin as boundary when viewed through the correspondence. Indeed, consider rays from the origin in ''Q'', and their images, vertical rays from the boundary ''R'' of ''HP''. Any point in ''HP'' is an infinite distance from the point ''p'' at the foot of the perpendicular to ''R'', but a sequence of points on this perpendicular may tend in the direction of ''p''. The corresponding sequence in ''Q'' tends along a ray toward the origin. The old Euclidean boundary of ''Q'' is no longer relevant. 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「Hyperbolic coordinates」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.