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'Euclidean distance'
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| Title of object: |
Euclidean distance |
| Canonical Name: |
EuclideanDistance |
| Type: |
Definition |
| Created on: |
2002-01-05 02:22:34 |
| Modified on: |
2004-04-28 10:12:32 |
| Classification: |
msc:54E35, msc:53A99 |
| Synonyms: |
Euclidean distance=Euclidean metric
Euclidean distance=standard metric
Euclidean distance=standard topology
Euclidean distance=Euclidean norm
Euclidean distance=Euclidean space
Euclidean distance=Euclidean
Euclidean distance=R^n |
Revision comment (for changes between this and next version):
| Changes for correction #4343 ('adding to "other names"'). |
Preamble:
%\usepackage{graphicx}
%\usepackage{xypic}
\usepackage{bbm}
\newcommand{\Z}{\mathbbmss{Z}}
\newcommand{\C}{\mathbbmss{C}}
\newcommand{\R}{\mathbbmss{R}}
\newcommand{\Q}{\mathbbmss{Q}}
\newcommand{\mathbb}[1]{\mathbbmss{#1}} |
Content:
If $u=(x_1,y_1)$ and $v=(x_2,y_2)$ are two points on the plane, their \emph{Euclidean distance} is given by
\begin{equation}\label{equno}
\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}.
\end{equation}
Geometrically, it's the length of the segment joining $u$ and $v$, and also the norm of the difference vector (considering $\R^n$ as vector space).
This distance induces a metric (and therefore a topology) on $\mathbb{R}^2$, called \emph{Euclidean metric (on $\R^2$)} or \emph{standard metric (on $\mathbb{R}^2)$}. The topology so induced is called \emph{standard topology} or \emph{usual topology on $\R^2$} and one basis can be obtained considering the set of all the open balls.
If $a=(x_1,x_2,\ldots,x_n)$ and $b=(y_1,y_2,\ldots,y_n)$, then formula \ref{equno} can be generalized to $\R^n$ by defining the Euclidean distance from $a$ to $b$ as
\begin{equation}d(a,b)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2+\cdots+(x_n-y_n)^2}.\end{equation}
Notice that this distance coincides with absolute value when $n=1$.
Euclidean distance on $\mathbb{R}^n$ is also a metric (Euclidean or standard metric), and therefore we can give $\mathbb{R}^n$ a topology, which is called the standard (canonical, usual, etc) topology of $\mathbb{R}^n$. The resulting (topological and vectorial) space is known as \emph{Euclidean space}.
This can also be done for $\C^n$ since as set $\C=\R^2$ and thus the metric on $\C$ is the same given to $\R^2$, and in general, $\C^n$ gets the same metric as $R^{2n}$. |
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