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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 Oriented projective geometry ： ウィキペディア英語版 Oriented projective geometry Oriented projective geometry is an oriented version of real projective geometry. Whereas the real projective plane describes the set of all unoriented lines through the origin in R3, the oriented projective plane describes lines with a given orientation. There are applications in computer graphics and computer vision where it is necessary to distinguish between rays light being emitted or absorbed by a point. Elements in an oriented projective space are defined using signed homogeneous coordinates. Let $\backslash mathbf\_^n$ be the set of elements of $\backslash mathbf^n$ excluding the origin. #Oriented projective line, $\backslash mathbf^1$: $(x,w)\; \backslash in\; \backslash mathbf^2\_$ *, with the equivalence relation $(x,w)\backslash sim(a\; x,a\; w)\backslash ,$ for all $a>0$. #Oriented projective plane, $\backslash mathbf^2$: $(x,y,w)\; \backslash in\; \backslash mathbf^3\_$ *, with $(x,y,w)\backslash sim(a\; x,a\; y,a\; w)\backslash ,$ for all $a>0$. These spaces can be viewed as extensions of euclidean space. $\backslash mathbf^1$ can be viewed as the union of two copies of $\backslash mathbf$, the sets (''x'',1) and (''x'',-1), plus two additional points at infinity, (1,0) and (-1,0). Likewise $\backslash mathbf^2$ can be view two copies of $\backslash mathbf^2$, (''x'',''y'',1) and (''x'',''y'',-1), plus one copy of $\backslash mathbf$ (''x'',''y'',0). An alternative way to view the spaces is as points on the circle or sphere, given by the points (''x'',''y'',''w'') with :''x''2+''y''2+''z''2=1. Distances between two points :$p=(p\_x,p\_y,p\_w)$ and :$q=(q\_x,q\_y,q\_w)$ in :$\backslash mathbf^2$ can be defined as elements in $\backslash mathbf^1$ :$((p\_x\; q\_w-q\_x\; p\_w)^2+(p\_y\; q\_w-q\_y\; p\_w)^2,\backslash mathrm(p\_w\; q\_w)(p\_w\; q\_w)^2)\backslash ,.$ ==References== * From original Stanford Ph.D. dissertation, ''Primitives for Computational Geometry'', available as (). * A. G. Oliveira, P. J. de Rezende, F. P. SelmiDei ''An Extension of CGAL to the Oriented Projective Plane T2 and its Dynamic Visualization System'', 21st Annual ACM Symp. on Computational Geometry, Pisa, Italy, 2005. * Nice introduction to oriented projective geometry in chapters 14 and 15. More at authors web-site. (Sherif Ghali ). 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「Oriented projective geometry」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.