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单词 ArcLength
释义

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

ab|F(t)|𝑑t=S

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

S=ab1+(f(x))2𝑑x

This formula is derived by viewing arclength as the Riemann sum

limΔxi=1n1+f(xi)2Δx

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

L=abr(θ)2+(r(θ))2𝑑θ
Titlearc lengthMathworldPlanetmath
Canonical nameArcLength
Date of creation2013-03-22 12:02:43
Last modified on2013-03-22 12:02:43
Ownermathcam (2727)
Last modified bymathcam (2727)
Numerical id14
Authormathcam (2727)
Entry typeAlgorithm
Classificationmsc 26B15
Synonymlength of a curve
Related topicRectifiable
Related topicIntegralRepresentationOfLengthOfSmoothCurve
Related topicStraightLineIsShortestCurveBetweenTwoPoints
Related topicPerimeterOfEllipse
Related topicEvolute2
Related topicCycloidMathworldPlanetmath
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