Length of a Curve

Abstract

Consider an arbitrary metric space M with the metric ρ. Let K be a curve in the space, and let ξ = (X₁, X₂, ..., X_m) be an arbitrary chain of the curve points, i.e., a finite sequence of the points of K, such that X₁ ≤ X₂ ≤ X_m. Let us set

$$ s\left( \xi \right) = \sum\limits_{i = 1}^{m = 1} {\rho \left( {{X_i},{X_{i + 1}}} \right)} . $$

. The least upper boundary of the quantity s(ξ) on the set of all chains of the curve K is called a length of the curve K and is denoted as s(K). The curve K is termed rectifiable if its length is finite.