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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 Zone diagram ： ウィキペディア英語版 Zone diagram A zone diagram is a certain geometric object which a variation on the notion of Voronoi diagram. It was introduced by Tetsuo Asano, Jiri Matousek, and Takeshi Tokuyama in 2007.〔 Formally, it is a fixed point of a certain function. Its existence or uniqueness are not clear in advance and have been established only in specific cases. Its computation is not obvious too. ==A particular but informative case== Consider a group of $n$ different pointuu $\backslash$ in the Euclidean plane. Each point is called a site. When we speak about the Voronoi diagram induced by these sites, we associate to the site $\backslash displaystyle$ the set $\backslash displaystyle$ of all points in the plane whose distance to the given site $\backslash displaystyle$ is not greater to their distance to any other site $p\_j,\backslash ,j\backslash neq\; k$. The collection $(R\_k)\_^n$ of these regions is the Voronoi diagram associated with these sites, and it induces a decomposition of the plane into regions: the Voronoi regions (Voronoi cells). In a zone diagram the region associated with the site $p\_k$ is defined a little bit differently: instead of associating it the set of all points whose distance to $p\_k$ is not greater than their distance to the other sites, we associate to $p\_k$ the set $R\_k$ of all points in the plane whose distance to $p\_k$ is not greater than their distance to any other region. Formally, :$R\_k=\backslash$. Here $\backslash displaystyle$ denotes the euclidean distance between the points $a$ and $b$ and $d(x,A)=\backslash inf\backslash$ is the distance between the point $x$ and the set $A$. In addition, $x=(x\_1,x\_2)\backslash in\; \backslash mathbb^2$ since we consider the plane. The tuple $(R\_k)\_^n$ is the zone diagram associated with the sites. The problem with this definition is that it seems circular: in order to know $R\_k$ we should know $\backslash displaystyle$ for each index $j,\backslash ,j\backslash neq\; k$ but each such $\backslash displaystyle$ is defined in terms of $\backslash displaystyle$. On a second thought, we see that actually the tuple $(R\_k)\_^n$ is a solution of the following system of equations: :$$ \begin R_1=\\, j\neq 1\}\\ \vdots\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\vdots\\ R_n=\\, j\neq n\} \end Rigorously, a zone diagram is any solution of this system, if such a solution exists. This definition can be extended without essentially any change to higher dimensions, to sites which are not necessarily points, to infinitely many sites, etc. 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「Zone diagram」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.