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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 Laguerre plane ： ウィキペディア英語版 Laguerre plane In mathematics, a Laguerre plane is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane, named after the French mathematician Edmond Nicolas Laguerre. Essentially the classical Laguerre plane is an incidence structure which describes the incidence behaviour of the curves $y=ax^2+bx+c$ , i.e. parabolas and lines, in the real affine plane. In order to simplify the structure, to any curve $y=ax^2+bx+c$ the point $(\backslash infty,a)$ is added. A further advantage of these completion is: The plane geometry of the completed parabolas/lines is isomorphic to the geometry of the plane sections of a cylinder (s. below). == The classical real Laguerre plane == Originally the classical Laguerre plane was defined as the geometry of the oriented lines and circles in the real euclidean plane (see ). Here we prefer the parabola model of the classical Laguerre plane. We define: $\backslash mathcal\; P:=\backslash R^2\backslash cup\; (\backslash \backslash times\backslash R),\; \backslash \; \backslash infty\; \backslash notin\; \backslash R,$ the set of points, $\backslash mathcal\; Z:=\backslash \; \backslash mid\; a,b,c\; \backslash in\; \backslash R\backslash \}$ the set of cycles. The incidence structure $(\backslash mathcal\; P,\backslash mathcal\; Z,\; \backslash in)$ is called classical Laguerre plane. The point set is $\backslash R^2$ plus a copy of $\backslash R$ (see figure). Any parabola/line $y=ax^2+bx+c$ gets the additional point $(\backslash infty,a)$. Points with the same x-coordinate cannot be connected by curves $y=ax^2+bx+c$. Hence we define: Two points $A,B$ are parallel ($A\backslash parallel\; B$) if $A=B$ or there is no cycle containing $A$ and $B$. For the description of the classical real Laguerre plane above two points $(a\_1,a\_2),\; (b\_1,b\_2)$ are parallel if and only if $a\_1=b\_1$. $\backslash parallel$ is an equivalence relation, similar to the parallelity of lines. The incidence structure $(\backslash mathcal\; P,\backslash mathcal\; Z,\; \backslash in)$ has the following properties: Lemma: : * For any three points $A,B,C$, pairwise not parallel, there is exactly one cycle $z$ containing $A,B,C$. : * For any point $P$ and any cycle $z$ there is exactly one point $P\text{'}\backslash in\; z$ such that $P\backslash parallel\; P\text{'}$. : * For any cycle $z$, any point $P\backslash in\; z$ and any point $Q\backslash notin\; z$ which is not parallel to $P$ there is exactly one cycle $z\text{'}$ through $P,Q$ with $z\backslash cap\; z\text{'}=\backslash$, i.e. $z$ and $z\text{'}$ touch each other at $P$''. Similar to the sphere model of the classical Moebius plane there is a cylinder model for the classical Laguerre plane: $(\backslash mathcal\; P,\backslash mathcal\; Z,\; \backslash in)$ is isomorphic to the geometry of plane sections of a circular cylinder in $\backslash R^3$ . The following mapping $\backslash Phi$ is a projection with center $(0,1,0)$ that maps the x-z-plane onto the cylinder with the equation $u^2+v^2-v=0$, axis $(0,\backslash tfrac,..)$ and radius $r=\backslash tfrac\backslash \; :$ :$\backslash Phi:\; \backslash \; (x,z)\; \backslash rightarrow\; (\backslash frac,\backslash frac,\backslash frac)=(u,v,w)\backslash \; .$ *The points $(0,1,a)$ (line on the cylinder through the center) apper not as images. *$\backslash Phi$ projects the ''parabola/line'' with equation $z=ax^2+bx+c$ into the plane $w-a=bu+(a-c)(v-1)$. So, the image of the parabola/line is the plane section of the cylinder with a non perpendicular plane and hence a circle/ellipse without point $(0,1,a)$. The parabolas/line $z=ax^2+a$ are mapped onto (horizontal) circles. *A line(a=0) is mapped onto a circle/Ellipse through center $(0,1,0)$ and a parabola ( $a\backslash ne0$) onto a circle/ellipse that do not contain $(0,1,0)$. 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「Laguerre plane」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.