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arc length (Algorithm)

Arclength is the length 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 simple, as it can be given by the sum of the lengths of the tangent vectors to the curve or

$\displaystyle \int_a^b \vert\vec{F}'(t)\vert \; dt=S $

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

$\displaystyle S=\int_a^b \sqrt{1+(f'(x))^2} \; dx $

This formula is derived by viewing arclength as the Riemann sum

$\displaystyle \lim_{\Delta x\rightarrow\infty}\sum_{i=1}^n \sqrt{1+f'(x_i)^2}\; \Delta x $

The term being summed is the length of an approximating secant to the curve over the distance $ \Delta x$. As $ \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

$\displaystyle L = \int_a^b \sqrt{r(\theta)^2 + (r'(\theta))^2}\; d\theta$



"arc length" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: rectifiable curve, integral representation of length of smooth curve, straight line is shortest curve between two points, perimeter of ellipse, evolute, cycloid

Other names:  length of a curve
Keywords:  curve length, length of an arc

Attachments:
perimeter of astroid (Example) by pahio
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Cross-references: polar coordinates, vanishes, secant to the curve, term, Riemann sum, Cartesian coordinates, parameter, tangent vectors, sum, equations, function, vector, distance, represent, applications, arc, length, curve, differentiable, section
There are 12 references to this entry.

This is version 10 of arc length, born on 2001-12-12, modified 2007-01-11.
Object id is 1090, canonical name is ArcLength.
Accessed 21478 times total.

Classification:
AMS MSC: 26B15 (Real functions :: Functions of several variables :: Integration: length, area, volume)
Pending Errata and Addenda
None.
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Discussion
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bad correction :-) by xriso on 2002-02-26 03:33:17
That correction was, of course, completely
wrong. In reality, it is L=\int_a^b \sqrt{r^2 + (dr/d\theta)^2} d\theta.
And those might have to be capital theta, I'm not sure.
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