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

Also known as
The Euclidean metric is also known as the Euclidean distance.

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

Also see

 * Euclidean Metric is Metric


 * Definition:Euclidean Norm
 * Definition:Product Metric