Euclidean Distance

Euclidean distance is a measure of the straight-line distance between any two points in a Euclidean space. Its quite literally just the Pythagorean theorem but scaled up quite a bit. Similar to cosine similarity, euclidean distance is often used to measure similarity within an embedding domain.

The general formula for Euclidean distance in $n$-dimensional space is

$$d(P,Q)=\sqrt{(p_1-q_1{)}^2+(p_2-q_2{)}^2+\cdots +(p_n-q_n{)}^2}$$

In one-dimensional space:

$$d(P,Q)=|p_1-q_1|$$

In two-dimensional space:

$$d(P,Q)=\sqrt{(p_1-q_1{)}^2+(p_2-q_2{)}^2}$$

In three-dimensional space:

$$d(P,Q)=\sqrt{(p_1-q_1{)}^2+(p_2-q_2{)}^2+(p_3-q_3{)}^2}$$

Euclidean distance has the following key properties:

1. It is non-negative $d(P,Q)\ge 0,\text{and }d(P,Q)=0\text{ if and only if }P=Q$
2. It is symmetric $d(P,Q)=d(Q,P)$
3. It has triangle inequality $d(P,Q)\le d(P,R)+d(R,Q)$