Euclidean distance

Remarks

The function $d$ is a metric on $\mathbb{R}^n$. For all points $p$, $q$ and $r$: $d(p, q) \geq 0$, $d(p, q) = 0$ if and only if $p = q$, $d(p, q) = d(q, p)$, and $d(p, r) \leq d(p, q) + d(q, r)$. For $n = 1$ the formula is $d(p, q) = |p - q|$. Some applications use the squared distance $d(p, q)^2$, because it has no square root. The squared distance is not a metric, because it does not satisfy the triangle inequality.