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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 Minkowski plane ： ウィキペディア英語版 Minkowski plane In mathematics, a Minkowski plane (named after Hermann Minkowski) is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane. ==Classical real Minkowski plane== Applying the pseudo-euclidean distance $d(P\_1,P\_2)=(x\text{'}\_1-x\text{'}\_2)^2-(y\text{'}\_1-y\text{'}\_2)^2$ on two points $P\_i=(x\text{'}\_i,y\text{'}\_i)$ (instead of the euclidean distance) we get the geometry of ''hyperbolas'', because a pseudo-euclidean circle $\backslash$ is a hyperbola with midpoint $M$. By a transformation of coordinates $x\_i=x\text{'}\_i+y\text{'}\_i$, $y\_i=x\text{'}\_i-y\text{'}\_i$, the pseudo-euclidean distance can be rewritten as $d(P\_1,P\_2)=(x\_1-x\_2)(y\_1-y\_2)$. The hyperbolas then have asymptotes parallel to the non-primed coordinate axes. The following completion (see Möbius and Laguerre planes) ''homogenizes'' the geometry of hyperbolas: : $\backslash mathcal\; P:=(\backslash R\backslash cup\; \backslash )^2=$ \R^2 \cup (\ \times\R) \cup (\R\times\) \ \cup \ \ , \ \infty \notin \R, the set of points, : $\backslash mathcal\; Z:=\backslash \; \backslash \; |$ \ a,b \in \R, a\ne 0\} ::: $\backslash cup\; \backslash +c,x\backslash ne\; b\backslash \}$ \cup \ \ | \ a,b,c \in \R, a\ne 0\}, the set of cycles. The incidence structure $(,,\backslash in)$ is called the classical real Minkowski plane. The set of points consists of $\backslash R^2$, two copies of $\backslash R$ and the point $(\backslash infty,\backslash infty)$. Any line $y=ax+b\; ,a\backslash ne0$ is completed by point $(\backslash infty,\backslash infty)$, any hyperbola $y=\backslash frac+c,a\backslash ne0$ by the two points $(b,\backslash infty),(\backslash infty,c)$ (see figure). Two points $(x\_1,y\_1)\backslash ne(x\_2,y\_2)$ can not be connected by a cycle if and only if $x\_1=x\_2$ or $y\_1=y\_2$. We define: Two points $P\_1,P\_2$ are (+)-parallel ($P\_1\backslash parallel\_+\; P\_2$) if $x\_1=x\_2$ and (−)-parallel ($P\_1\backslash parallel\_-\; P\_2$) if $y\_1=y\_2$. Both these relations are equivalence relations on the set of points. Two points $P\_1,P\_2$ are called parallel ($P\_1\backslash parallel\; P\_2$) if $P\_1\backslash parallel\_+\; P\_2$ or $P\_1\backslash parallel\_-\; P\_2$. From the definition above we find: Lemma: : *For any pair of non parallel points $A,B$ there is exactly one point $C$ with $A\backslash parallel\_+\; C\; \backslash parallel\_-\; B$. : *For any point $P$ and any cycle $z$ there are exactly two points $A,B\; \backslash in\; z$ with $A\backslash parallel\_+\; P\; \backslash parallel\_-\; B$. : *For any three points $A$, $B$, $C$, pairwise non parallel, there is exactly one cycle $z$ that contains $A,B,C$. : *For any cycle $z$, any point $P\backslash in\; z$ and any point $Q,\; P\; \backslash not\backslash parallel\; Q$ and $Q\backslash notin\; z$ there exists exactly one cycle $z\text{'}$ such that $z\backslash cap\; z\text{'}=\backslash$, i.e. $z$ touches $z\text{'}$ at point P. Like the classical Möbius and Laguerre planes Minkowski planes can be described as the geometry of plane sections of a suitable quadric. But in this case the quadric lives in projective 3-space: The classical real Minkowski plane is isomorphic to the geometry of plane sections of a hyperboloid of one sheet (not degenerated quadric of index 2). 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「Minkowski 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.