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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 Line–plane intersection ： ウィキペディア英語版 Line–plane intersection In analytic geometry, the intersection of a line and a plane can be the empty set, a point, or a line. Distinguishing these cases, and determining equations for the point and line in the latter cases, have use in computer graphics, motion planning, and collision detection. ==Algebraic form== In vector notation, a plane can be expressed as the set of points $\backslash mathbf$ for which :$(\backslash mathbf-\backslash mathbf)\backslash cdot\backslash mathbf\; =\; 0$ where $\backslash mathbf$ is a normal vector to the plane and $\backslash mathbf$ is a point on the plane. (The notation $\backslash mathbf\backslash cdot\backslash mathbf$ denotes the dot product of the two vector $\backslash mathbf$ and $\backslash mathbf$.) The vector equation for a line is :$\backslash mathbf\; =\; d\backslash mathbf\; +\; \backslash mathbf\; \backslash quad\; d\backslash in\backslash mathbb$ where $\backslash mathbf$ is a vector in the direction of the line, $\backslash mathbf$ is a point on the line, and $d$ is a scalar in the real number domain. Substitute the equation for the line into the equation for the plane gives :$(d\; \backslash mathbf\; +\; \backslash mathbf\; -\; \backslash mathbf)\backslash cdot\backslash mathbf\; =\; 0$ Expanding gives :$d\; \backslash mathbf\backslash cdot\backslash mathbf\; +\; (\backslash mathbf-\backslash mathbf)\backslash cdot\backslash mathbf\; =\; 0$ And solve for $d$ :$d\; =\; )\backslash cdot\backslash mathbf\; \backslash over\; \backslash mathbf\backslash cdot\backslash mathbf\}.$ If $\backslash mathbf\backslash cdot\backslash mathbf\; =\; 0$ then the line and plane are parallel. There will be two cases: if $(\backslash mathbf-\backslash mathbf)\backslash cdot\backslash mathbf\; =0$ then the line is contained in the plane, that is, the line intersects the plane at each point of the line. Otherwise, the line and plane have no intersection. If $\backslash mathbf\backslash cdot\backslash mathbf\; \backslash ne\; 0$ there is a single point of intersection. The value of $d$ can be calculated and the point of intersection is given by :$d\backslash mathbf\; +\; \backslash mathbf$. 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「Line–plane intersection」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.