arc length

Arclength is the of a sectionMathworldPlanetmath of a differentiableMathworldPlanetmathPlanetmath curve. Finding the length of an arc is useful in many applications, for the length of a curve can represent distance traveled, work, etc. It is commonly represented as S or the differential ds if one is differentiating or integrating with respect to change in arclength.

If one knows the vector function or parametric equations of a curve, finding the arclength is , as it can be given by the sum of the lengths of the tangent vectorsMathworldPlanetmath to the curve or


Note that t is an independent parameter. In Cartesian coordinatesMathworldPlanetmath, arclength can be calculated by the formula


This formula is derived by viewing arclength as the Riemann sum


The term being summed is the length of an approximating secant to the curve over the distance Δx. As Δx vanishes, the sum approaches the arclength, as desired. Arclength can also be derived for polar coordinatesMathworldPlanetmath from the general formula for vector functions given above. The result is

Title arc lengthMathworldPlanetmath
Canonical name ArcLength
Date of creation 2013-03-22 12:02:43
Last modified on 2013-03-22 12:02:43
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 14
Author mathcam (2727)
Entry type Algorithm
Classification msc 26B15
Synonym length of a curve
Related topic Rectifiable
Related topic IntegralRepresentationOfLengthOfSmoothCurve
Related topic StraightLineIsShortestCurveBetweenTwoPoints
Related topic PerimeterOfEllipse
Related topic Evolute2
Related topic CycloidMathworldPlanetmath