Euclidean Distance

geometry

Definition

Euclidean Distance

The Euclidean distance (or $L_2$ distance) is the straight-line distance between two points in Euclidean space. Formally, for two vectors $\mathbf{x},\mathbf{v}\in {\mathbb{R}}^m$, the Euclidean distance $d(\mathbf{x},\mathbf{v})$ is defined as the $L_2$ norm of their difference:

$$d(\mathbf{x},\mathbf{v})=\Vert \mathbf{x}-\mathbf{v}{\Vert }_2=\sqrt{\sum _{i=1}^m(x_i-v_i{)}^2}$$

Properties

Rotational Invariance: The Euclidean distance remains invariant under any rotation or translation of the coordinate system, preserving the geometric relationship between points.

Error Penalisation: Compared to the Manhattan distance, the squaring operation in the $L_2$ norm ensures that larger coordinate differences contribute disproportionately to the total distance.