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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–sphere intersection ： ウィキペディア英語版 Line–sphere intersection In analytic geometry, a line and a sphere can intersect in three ways: no intersection at all, at exactly one point, or in two points. Methods for distinguishing these cases, and determining equations for the points in the latter cases, are useful in a number of circumstances. For example, this is a common calculation to perform during ray tracing (Eberly 2006:698). == Calculation using vectors in 3D == In vector notation, the equations are as follows: Equation for a sphere :$\backslash left\backslash Vert\; \backslash mathbf\; -\; \backslash mathbf\; \backslash right\backslash Vert^2=r^2$ : *$\backslash mathbf$ - center point : *$r$ - radius : *$\backslash mathbf$ - points on the sphere Equation for a line starting at $\backslash mathbf$ :$\backslash mathbf=\backslash mathbf\; +\; d\backslash mathbf$ : *$d$ - distance along line from starting point : *$\backslash mathbf$ - direction of line (a unit vector) : *$\backslash mathbf$ - origin of the line : *$\backslash mathbf$ - points on the line Searching for points that are on the line and on the sphere means combining the equations and solving for $d$: :Equations combined ::$\backslash left\backslash Vert\; \backslash mathbf\; +\; d\backslash mathbf\; -\; \backslash mathbf\; \backslash right\backslash Vert^2=r^2\; \backslash Leftrightarrow\; (\backslash mathbf\; +\; d\backslash mathbf\; -\; \backslash mathbf)\; \backslash cdot\; (\backslash mathbf\; +\; d\backslash mathbf\; -\; \backslash mathbf)\; =\; r^2$ :Expanded ::$d^2(\backslash mathbf\backslash cdot\backslash mathbf)+2d(\backslash mathbf\backslash cdot(\backslash mathbf-\backslash mathbf))+(\backslash mathbf-\backslash mathbf)\backslash cdot(\backslash mathbf-\backslash mathbf)=r^2$ :Rearranged ::$d^2(\backslash mathbf\backslash cdot\backslash mathbf)+2d(\backslash mathbf\backslash cdot(\backslash mathbf-\backslash mathbf))+(\backslash mathbf-\backslash mathbf)\backslash cdot(\backslash mathbf-\backslash mathbf)-r^2=0$ :The form of a quadratic formula is now observable. (This quadratic equation is an example of Joachimsthal's Equation ().) ::$a\; d^2\; +\; b\; d\; +\; c\; =\; 0$ :where : *$a=\backslash mathbf\backslash cdot\backslash mathbf=\backslash left\backslash Vert\backslash mathbf\backslash right\backslash Vert^2$ : *$b=2(\backslash mathbf\backslash cdot(\backslash mathbf-\backslash mathbf))$ : *$c=(\backslash mathbf-\backslash mathbf)\backslash cdot(\backslash mathbf-\backslash mathbf)-r^2=\backslash left\backslash Vert\backslash mathbf-\backslash mathbf\backslash right\backslash Vert^2-r^2$ :Simplified ::$d=\backslash frac-\backslash mathbf))\; \backslash pm\; \backslash sqrt-\backslash mathbf))^2-\backslash left\backslash Vert\backslash mathbf\backslash right\backslash Vert^2(\backslash left\backslash Vert\backslash mathbf-\backslash mathbf\backslash right\backslash Vert^2-r^2)\}\}$ :Note that $\backslash mathbf$ is a unit vector, and thus $\backslash left\backslash Vert\backslash mathbf\backslash right\backslash Vert^2=1$. Thus, we can simplify this further to ::$d=-(\backslash mathbf\backslash cdot(\backslash mathbf-\backslash mathbf))\; \backslash pm\; \backslash sqrt-\backslash mathbf))^2-\backslash left\backslash Vert\backslash mathbf-\backslash mathbf\backslash right\backslash Vert^2+r^2\}$ *If the value under the square-root ($(\backslash mathbf\backslash cdot(\backslash mathbf-\backslash mathbf))^2-\backslash left\backslash Vert\backslash mathbf-\backslash mathbf\backslash right\backslash Vert^2+r^2$) is less than zero, then it is clear that no solutions exist, i.e. the line does not intersect the sphere (case 1). *If it is zero, then exactly one solution exists, i.e. the line just touches the sphere in one point (case 2). *If it is greater than zero, two solutions exist, and thus the line touches the sphere in two points (case 3). 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–sphere 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.