Definition:Euclidean Metric/General Definition
Definition
Let $M_{1'} = \struct {A_{1'}, d_{1'} }, M_{2'} = \struct {A_{2'}, d_{2'} }, \ldots, M_{n'} = \struct {A_{n'}, d_{n'} }$ be metric spaces.
Let $\ds \AA = \prod_{i \mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \ldots, A_{n'}$.
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
Real Vector Space
Let $\ds \AA = \prod_{i \mathop = 1}^n A_{i'}$ be the real vector space $\R^n$.
The Euclidean metric on $\R^n$ is defined as:
- $\ds \map {d_2} {x, y} := \paren {\sum_{i \mathop = 1}^n \paren {x_i - y_i}^2}^{1 / 2}$
where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \R^n$.
Real Number Line
On the real number line, the Euclidean metric can be seen to degenerate to:
- $\map d {x, y} := \sqrt {\paren {x - y}^2} = \size {x - y}$
where $\size {x - y}$ denotes the absolute value of $x - y$.
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 $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \R^2$.
Ordinary Space
The Euclidean metric on $\R^3$ is defined as:
- $\map {d_2} {x, y} := \sqrt {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2 + \paren {x_3 - y_3}^2}$
where $x = \tuple {x_1, x_2, x_3}, y = \tuple {y_1, y_2, y_3} \in \R^3$.
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 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, 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 $2.2.7$