Definition:Euclidean Metric

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

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:

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

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

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

Euclidean Metric is Metric

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

1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): $\text{III}$: Pythagoras' Theorem
1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.7$
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: Euclidean distance or Cartesian distance
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: Euclidean metric
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: Euclidean distance (Cartesian distance)