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 $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \AA$.
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:
- $\forall z_1, z_2 \in \C: \map d {z_1, z_2} := \size {z_1 - z_2}$
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 call it the product metric.
Some sources refer to it as the Cartesian distance or Cartesian metric, for René Descartes.
The Euclidean metric is sometimes also referred to as the usual metric.
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
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: 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): Euclidean distance or Cartesian distance
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Riemannian geometry
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cartesian metric
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Euclidean metric
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Riemannian geometry
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Euclidean distance (Cartesian distance)