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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 Hesse normal form ： ウィキペディア英語版 Hesse normal form The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in $\backslash mathbb^2$ or a plane in Euclidean space $\backslash mathbb^3$ or a hyperplane in higher dimensions.〔.〕 It is primarily used for calculating distances, and is written in vector notation as :$\backslash vec\; r\; \backslash cdot\; \backslash vec\; n\_0\; -\; d\; =\; 0.\backslash ,$ This equation is satisfied by all points ''P'' described by the location vector $\backslash vec\; r$, which lie precisely in the plane ''E'' (or in 2D, on the line ''g''). The vector $\backslash vec\; n\_0$ represents the unit normal vector of ''E'' or ''g'', that points from the origin of the coordinate system to the plane (or line, in 2D). The distance $d\; \backslash ge\; 0$ is the distance from the origin to the plane (or line). The dot $\backslash cdot$ indicates the scalar product or dot product. == Derivation/Calculation from the normal form == Note: For simplicity, the following derivation discusses the 3D case. However, it is also applicable in 2D. In the normal form, :$(\backslash vec\; r\; -\backslash vec\; a)\backslash cdot\; \backslash vec\; n\; =\; 0\backslash ,$ a plane is given by a normal vector $\backslash vec\; n$ as well as an arbitrary position vector $\backslash vec\; a$ of a point $A\; \backslash in\; E$. The direction of $\backslash vec\; n$ is chosen to satisfy the following inequality :$\backslash vec\; a\backslash cdot\; \backslash vec\; n\; \backslash geq\; 0\backslash ,$ By dividing the normal vector $\backslash vec\; n$ by its Magnitude $|\; \backslash vec\; n\; |$, we obtain the unit (or normalized) normal vector :$\backslash vec\; n\_0\; =\; \backslash ,$ and the above equation can be rewritten as :$(\backslash vec\; r\; -\backslash vec\; a)\backslash cdot\; \backslash vec\; n\_0\; =\; 0.\backslash ,$ Substituting :$d\; =\; \backslash vec\; a\backslash cdot\; \backslash vec\; n\_0\; \backslash geq\; 0\backslash ,$ we obtain the Hesse normal form :$\backslash vec\; r\; \backslash cdot\; \backslash vec\; n\_0\; -\; d\; =\; 0.\backslash ,$ File:Ebene Hessesche Normalform.PNG In this diagram, ''d'' is the distance from the origin. Because $\backslash vec\; r\; \backslash cdot\; \backslash vec\; n\_0\; =\; d$ holds for every point in the plane, it is also true at point ''Q'' (the point where the vector from the origin meets the plane E), with $\backslash vec\; r\; =\; \backslash vec\; r\_s$, per the definition of the Scalar product :$d\; =\; \backslash vec\; r\_s\; \backslash cdot\; \backslash vec\; n\_0\; =\; |\backslash vec\; r\_s|\; \backslash cdot\; |\backslash vec\; n\_0|\; \backslash cdot\; \backslash cos(0^\backslash circ)\; =\; |\backslash vec\; r\_s|\; \backslash cdot\; 1\; =\; |\backslash vec\; r\_s|.\backslash ,$ The magnitude $|\backslash vec\; r\_s|$ of $$ is the shortest distance from the origin to the plane. 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「Hesse normal form」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.