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 problem in vector calculus and in differential geometry. In the most basic formulation of arc length for a vector valued curve (thought of as the trajectory of a particle), the arc length is obtained by integrating the the magnitude of the velocity vector over the curve with respect to time. 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).