Arc length

Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a focus of calculus. In the most basic formulation of arc length for a parametric curve (thought of as the trajectory of a particle), the arc length is gotten by integrating the speed of the particle over the path. Thus the length of a continuously differentiable curve ${\displaystyle (x(t),y(t))}$, for ${\displaystyle a\leq t\leq b}$, in the Euclidean plane is given as the integral $${\displaystyle L=\int _{a}^{b}{\sqrt {x'(t)^{2}+y'(t)^{2}}}\,dt,}$$ (because ${\displaystyle {\sqrt {x'(t)^{2}+y'(t)^{2}}}}$ is the magnitude of the velocity vector ${\displaystyle (x(t),y(t))}$, i.e., the particle's speed).

The defining integral of arc length does not always have a closed-form expression, and numerical integration may be used instead to obtain numerical values of arc length.

Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is also called curve rectification. For a rectifiable curve these approximations don't get arbitrarily large (so the curve has a finite length).