Euclidean Distance
The Euclidean distance is the most commonly used geometric distance metric, it returns the distance between two points on a plane.
Mathematically, the Euclidean distance is given as:
$$ d(x, y) = \sqrt { \sum_{i=1}^n (x_i - y_i)^2 } $$
where $x$ and $y$ are vectors.
The expanded form of the equation is for two points in a 2-D vector space is:
$$ d(x, y) = \sqrt { (x_1 - x_2)^2 + (y_1 - y_2)^2 } $$
For example, suppose we have two points:
A = (6.3, 4.1), B = (3.2, 1.7)
The Euclidean distance between the two points is given by:
$$ d(A,B) = \sqrt { (6.3 - 3.2)^2 + (4.1 - 1.7)^2 } $$ $$ d(A,B) = \sqrt { (3.1)^2 + (2.4)^2 }$$ $$ d(A, B) = \sqrt { (9.6) + (5.76) } = \sqrt { 15.369 } = 3.92 $$
Implementation with python code:
import numpy as np
a = np.array([6.3, 4.1])
b = np.array([3.2, 1.7])
d = np.sqrt( np.sum( (a - b)**2 ) )
d