arc length

Arclength is the of a section of a differentiable 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 vectors to the curve or

$${\int }_a^b|{\overrightarrow{F}}^{\prime }(t)|𝑑t=S$$

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

$$S={\int }_a^b\sqrt{1+{(f^{\prime }(x))}^2}𝑑x$$

This formula is derived by viewing arclength as the Riemann sum

$$\underset{\mathrm{\Delta }x\to \mathrm{\infty }}{lim}\sum _{i=1}^n\sqrt{1+f^{\prime }{(x_i)}^2}\mathrm{\Delta }x$$

The term being summed is the length of an approximating secant to the curve over the distance $\mathrm{\Delta }x$. As $\mathrm{\Delta }x$ vanishes, the sum approaches the arclength, as desired. Arclength can also be derived for polar coordinates from the general formula for vector functions given above. The result is

$$L={\int }_a^b\sqrt{r{(\theta )}^2+{(r^{\prime }(\theta ))}^2}𝑑\theta$$

Title	arc length
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	Cycloid