Definition:Euclidean Metric/General Definition

From ProofWiki
Jump to navigation Jump to search

Definition

Let $M_{1'} = \left({A_{1'}, d_{1'}}\right), M_{2'} = \left({A_{2'}, d_{2'}}\right), \ldots, M_{n'} = \left({A_{n'}, d_{n'}}\right)$ be metric spaces.

Let $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \ldots, A_{n'}$.


The Euclidean metric on $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ is defined as:

$\displaystyle d_2 \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left({d_{i'} \left({x_i, y_i}\right)}\right)^2}\right)^{\frac 1 2}$

where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$.


Special Cases

Real Vector Space

Let $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ be the real vector space $\R^n$.


The Euclidean metric on $\R^n$ is defined as:

$\displaystyle d_2 \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left({x_i - y_i}\right)^2}\right)^{1 / 2}$

where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \R^n$.


Real Number Line

On the real number line, the Euclidean metric can be seen to degenerate to:

$d \left({x, y}\right) := \sqrt {\left({x - y}\right)^2} = \left|{x - y}\right|$

where $\left|{x - y}\right|$ denotes the absolute value of $x - y$.


Real Number Plane

The Euclidean metric on $\R^2$ is defined as:

$\displaystyle d_2 \left({x, y}\right) := \sqrt{\left({x_1 - y_1}\right)^2 + \left({x_2 - y_2}\right)^2}$

where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in \R^2$.


Ordinary Space

The Euclidean metric on $\R^3$ is defined as:

$\displaystyle d_2 \left({x, y}\right) := \sqrt{\left({x_1 - y_1}\right)^2 + \left({x_2 - y_2}\right)^2 + \left({x_3 - y_3}\right)^2}$

where $x = \left({x_1, x_2, x_3}\right), y = \left({y_1, y_2, y_3}\right) \in \R^3$.


Complex Plane

The Euclidean metric on $\C$ is defined as:

$\displaystyle \forall z_1, z_2 \in \C: d \left({z_1, z_2}\right) := \left\vert{z_1 - z_2}\right\vert$

where $\left\vert{z_1 - z_2}\right\vert$ denotes the modulus of $z_1 - z_2$.


Also see


Source of Name

This entry was named for Euclid.


Historical Note

Euclid himself did not in fact conceive of the Euclidean metric and its associated Euclidean Space and Euclidean norm.

They bear that name because the geometric space which it gives rise to is Euclidean in the sense that it is consistent with Euclid's fifth postulate.


Sources