Two points pi, qi

# Euclidean distance

d(pi,qi) = (∑(pi-qi)^2)^(1/2)

# Chebyshev distance

measures distance assuming only the most significant dimension is relevant.

d(pi,qi) = max(|pi-qi|) max for each i

Also called L∞ metric, as d = lim p->∞ (∑|pi-qi|^p)^(1/p)

# Manhattan distance

d(pi,qi) = ∑i (|pi-qi|)

measures distance following only axis-aligned directions.

# Minkowski distance of order p (p-norm distance)

d =  (∑|pi-qi|^p)^(1/p)

a generalization that unifies Euclidean distance (p=2), Manhattan distance (p=1), and Chebyshev distance (p=∞).