arclength

Arc length, also written as arclength, is a measure of the length of a curve in Euclidean space. For a differentiable curve γ: [a, b] → R^n, the arc length L is defined by L = ∫_a^b ||γ′(t)|| dt, where ||·|| denotes the Euclidean norm. The differential form ds = ||dγ|| = sqrt(dx^2 + dy^2 + dz^2) expresses the infinitesimal arc length along the curve.

In the plane, if a curve is given as the graph y = f(x) with x ∈ [a, b], the arc length is L = ∫_a^b sqrt(1 + (f′(x))^2) dx. For a parametric plane curve γ(t) = (x(t), y(t)) with t ∈ [a, b], L = ∫_a^b sqrt((dx/dt)^2 + (dy/dt)^2) dt. In three dimensions, a space curve γ(t) = (x(t), y(t), z(t)) has L = ∫_a^b sqrt((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2) dt.

Arc length is invariant under reparameterization: the length depends only on the traced curve, not on how it is parameterized. If s is defined as the accumulated arc length from some base parameter, s(t) = ∫_{t0}^t ||γ′(u)|| du, one can reparameterize the curve by arc length to obtain a unit-speed (or unit-energy) parameterization γ̃(s) with ||γ̃′(s)|| = 1.

Special cases include straight line segments, where L equals the Euclidean distance between endpoints, and full circular arcs, where L = Rθ for a circle of radius R and central angle θ. Arc length is a fundamental concept in physics, engineering, computer graphics, and differential geometry, enabling measurement of travel distance, curve comparison, and distance along curves.