Definition:Euclidean Metric

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Definition

Let $M_{1'} = \struct {A_{1'}, d_{1'} }$ and $M_{2'} = \struct {A_{2'}, d_{2'} }$ be metric spaces.

Let $A_{1'} \times A_{2'}$ be the cartesian product of $A_{1'}$ and $A_{2'}$.


The Euclidean metric on $A_{1'} \times A_{2'}$ is defined as:

$\map {d_2} {x, y} := \paren {\paren {\map {d_{1'} } {x_1, y_1} }^2 + \paren {\map {d_{2'} } {x_2, y_2} }^2}^{1/2}$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in A_{1'} \times A_{2'}$.


General Definition

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

$\ds \map {d_2} {x, y} := \paren {\sum_{i \mathop = 1}^n \paren {\map {d_{i'} } {x_i, y_i} }^2}^{\frac 1 2}$

where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \AA$.


Riemannian Manifold

Let $x \in \R^n$ be a point.

Let $\tuple {x_1, \ldots, x_n}$ be the standard coordinates.

Let $T_x \R^n$ be the tangent space of $\R^n$ at $x$.

Let $T_x \R^n$ be identified with $\R^n$:

$T_x \R^n \cong \R^n$


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Let $v, w \in T_x \R^n$ be vectors such that:

$\ds v = \sum_{i \mathop = 1}^n v^i \valueat {\partial_i} x$
$\ds w = \sum_{i \mathop = 1}^n w^i \valueat {\partial_i} x$

Let $g$ be a Riemannian metric such that:

$\ds g_x = \innerprod v w_x = \sum_{i \mathop = 1}^n v^i w^i$


Then $g$ is called the Euclidean metric.


Special Cases


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Real Number Plane

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

$\ds \map {d_2} {x, y} := \sqrt {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2}$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \R^2$.


Rational Number Plane

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

$\ds \map {d_2} {x, y} := \sqrt {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2}$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \Q^2$.


Complex Plane

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

$\d \forall z_1, z_2 \in \C: \map d {z_1, z_2} := \size {z_1 - z_2}$

where $\size {z_1 - z_2}$ denotes the modulus of $z_1 - z_2$.


Also known as

The Euclidean metric is also known as the Euclidean distance.

Some sources refer to it as the Cartesian distance, for René Descartes.


Also see

  • Results about the Euclidean metric can be found here.


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, Euclidean topology 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

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