View source for Arc length

{{Short description|Distance along a curve}}
[[File:Arc length.gif|right|frame|When rectified, the curve gives a straight line segment with the same length as the curve's arc length.]]
[[File:Logarithmic spiral arc length.gif|thumb|right|Arc length ''s'' of a [[logarithmic spiral]] as a function of its parameter ''θ''.]]

'''Arc length''' is the distance between two points along a section of a [[curve]].

Determining the length of an irregular arc segment is also called {{em|rectification}} of a curve. The advent of [[infinitesimal calculus]] led to a general formula that provides [[closed-form expression|closed-form solutions]] in some cases.

== General approach ==
[[File:Arclength.svg|400px|right|thumb|Approximation by multiple linear segments]]

A [[curve]] in the [[Euclidean space|plane]] can be approximated by connecting a [[wiktionary:Finite|finite]] number of [[point (geometry)|points]] on the curve using [[line segment]]s to create a [[polygonal chain|polygonal path]]. Since it is straightforward to calculate the [[length]] of each linear segment (using the [[Pythagorean theorem]] in Euclidean space, for example), the total length of the approximation can be found by [[summation|summing]] the lengths of each linear segment; {{anchor|Chordal distance}}that approximation is known as the ''(cumulative) [[chord (geometry)|chordal]] distance''.<ref>{{cite book|page=51|last1=Ahlberg|last2=Nilson|date=1967|title=The Theory of Splines and Their Applications|publisher=Academic Press|url=https://books.google.com/books?id=S7d1pjJHsRgC&lpg=PA51|isbn=9780080955452}}</ref>

If the curve is not already a polygonal path, using a progressively larger number of segments of smaller lengths will result in better approximations. The lengths of the successive approximations will not decrease and may keep increasing indefinitely, but for smooth curves they will tend to a finite limit as the lengths of the segments get [[arbitrarily large|arbitrarily small]].

For some curves there is a smallest number <math>L</math> that is an upper bound on the length of any polygonal approximation. These curves are called {{em|rectifiable}} and the number <math>L</math> is defined as the {{em|arc length}}.

==Definition for a smooth curve==
{{See also|Curve#Length_of_a_curve|l1=Length of a curve}}

Let <math>f\colon[a,b]\to\R^n</math> be a [[Continuous function|continuously]] [[Differentiable function|differentiable]] function.  The length of the curve defined by <math>f</math> can be defined as the [[Limit (mathematics)|limit]] of the sum of line segment lengths for a regular partition of <math>[a,b]</math> as the number of segments approaches infinity.  This means

:<math>L(f)=\lim_{N\to\infty}\sum_{i=1}^N \bigg|f(t_i)-f(t_{i-1})\bigg|</math>

where <math>t_i=a+i(b-a)/N=a+i\Delta t</math> for <math>i=0,1,\dotsc,N.</math> This definition is equivalent to the standard definition of arc length as an integral:

:<math>\lim_{N\to\infty}\sum_{i=1}^N \bigg|f(t_i)-f(t_{i-1})\bigg|=\lim_{N\to\infty}\sum_{i=1}^N \left|\frac{f(t_i)-f(t_{i-1})}{\Delta t}\right|\Delta t=\int_a^b \Big|f'(t)\Big|\ dt.</math>

The last equality above is true because the definition of the derivative as a limit implies that there is a positive real function <math>\delta(\epsilon)</math>  of positive real <math>\epsilon</math> such that <math>\Delta t<\delta(\epsilon)</math> implies <math>\left|\bigg|\frac{f(t_i)-f(t_{i-1})}{\Delta t}\bigg|-\Big|f'(t_i)\Big|\right|<\epsilon.</math> This means

:<math>
\sum_{i=1}^N \left|\frac{f(t_i)-f(t_{i-1})}{\Delta t}\right|\Delta t-\sum_{i=1}^N \Big|f'(t_i)\Big|\Delta t
</math>

has absolute value less than <math>\epsilon (b-a)</math> for <math>N>(b-a)/\delta(\epsilon).</math>  This means that in the limit <math>N\rightarrow\infty,</math> the left term above equals the right term, which is just the [[Riemann integral]] of <math>|f'(t)|</math> on  <math>[a,b].</math>  This definition of arc length shows that the length of a curve <math>f:[a,b]\rightarrow\mathbb{R}^n</math> continuously differentiable on <math>[a,b]</math> is always finite.  In other words, the curve is always rectifiable.

The definition of arc length of a smooth curve as the integral of the norm of the derivative is equivalent to the definition

:<math>L(f)=\sup\sum_{i=1}^N \bigg|f(t_i)-f(t_{i-1})\bigg|</math>

where the [[Infimum and supremum|supremum]] is taken over all possible partitions <math>a=t_0<t_1<\dots <t_{N-1}<t_N=b</math> of <math>[a,b].</math><ref>{{Cite book|title = Principles of Mathematical Analysis|last = Rudin|first = Walter|publisher = McGraw-Hill, Inc.|year = 1976|isbn = 978-0-07-054235-8|location = |pages = 137}}</ref> This definition is also valid if <math>f</math> is merely continuous, not differentiable.

A curve can be parameterized in infinitely many ways. Let <math>\varphi:[a,b]\to [c,d]</math> be any continuously differentiable [[bijection]]. Then <math>g=f\circ\varphi^{-1}:[c,d]\to\R^n</math> is another continuously differentiable parameterization of the curve originally defined by <math>f.</math> The arc length of the curve is the same regardless of the parameterization used to define the curve:

:<math>\begin{align}
L(f) &= \int_a^b \Big|f'(t)\Big|\ dt=\int_a^b \Big|g'(\varphi(t))\varphi'(t)\Big|\ dt \\
&= \int_a^b \Big|g'(\varphi(t))\Big|\varphi'(t)\ dt \quad \textrm{since} \ \varphi \ \textrm{must} \ \textrm{be} \ \textrm{non-decreasing} \\
&= \int_c^d \Big|g'(u)\Big|\ du \quad \textrm{using} \ \textrm{integration} \ \textrm{by} \ \textrm{substitution}\\
&= L(g).
\end{align}</math>

==Finding arc lengths by integrating==
{{See also|Differential geometry of curves#Length and natural parametrization|l1=Differential geometry of curves}}

[[File:Quarter circle.png|thumb|400x400px|Quarter circle]]
If a [[plane curve|planar curve]] in <math>\mathbb{R}^2</math> is defined by the equation <math> y=f(x), </math> where <math>f</math> is continuously differentiable, then it is simply a special case of a parametric equation where <math>x = t </math> and <math> y = f(t).</math>  The arc length is then given by:

:<math>s=\int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}dx.</math>

Curves with [[Solution in closed form|closed-form solutions]] for arc length include the [[catenary]], [[circle]], [[cycloid]], [[logarithmic spiral]], [[parabola]], [[semicubical parabola]] and [[line (mathematics)|straight line]]. The lack of a closed form solution for the arc length of an [[Ellipse#Circumference|elliptic]] and [[Hyperbola|hyperbolic]] arc led to the development of the [[elliptic integral]]s.

===Numerical integration===
In most cases, including even simple curves, there are no closed-form solutions for arc length and [[numerical integration]] is necessary.  Numerical integration of the arc length integral is usually very efficient.  For example, consider the problem of finding the length of a quarter of the unit circle by numerically integrating the arc length integral.  The upper half of the unit circle can be parameterized as <math>y=\sqrt{1-x^2}.</math>  The interval <math>x\in [-\sqrt{2}/2, \sqrt{2}/2] </math> corresponds to a quarter of the circle.  Since <math>dy/dx=-x/\sqrt{1-x^2}</math> and <math>1+(dy/dx)^2 = 1/(1-x^2),</math> the length of a quarter of the unit circle is

:<math>\int_{-\sqrt{2}/2}^{\sqrt{2}/2}\frac{1}{\sqrt{1-x^2}} \, dx.</math>

The 15-point [[Gauss–Kronrod quadrature formula|Gauss–Kronrod]] rule estimate for this integral of {{val|1.570796326808177}} differs from the true length of

:<math>\Big[\arcsin x\Big]^{\sqrt{2}/2}_{-\sqrt{2}/2}=\frac \pi 2</math>

by {{val|1.3e-11}} and the 16-point [[Gaussian quadrature]] rule estimate of {{val|1.570796326794727}} differs from the true length by only {{val|1.7e-13}}.  This means it is possible to evaluate this integral to almost [[Machine epsilon|machine precision]] with only 16 integrand evaluations.

=== Curve on a surface ===
Let <math>\mathbf{x}(u,v)</math> be a surface mapping and let <math>\mathbf{C}(t) = (u(t), v(t))</math> be a curve on this surface.  The integrand of the arc length integral is <math>|(\mathbf{x}\circ\mathbf{C})'(t)|.</math>  Evaluating the derivative requires the [[chain rule]] for vector fields:

: <math>D(\mathbf{x} \circ \mathbf{C}) = (\mathbf{x}_u \ \mathbf{x}_v)\binom{u'}{v'} = \mathbf{x}_u u' + \mathbf{x}_v v'.</math>

The squared norm of this vector is <math>(\mathbf{x}_u u' + \mathbf{x}_v v') \cdot (\mathbf{x}_u u' + \mathbf{x}_v v') = g_{11}(u')^2 + 2g_{12}u'v' + g_{22}(v')^2 </math> (where <math>g_{ij} </math> is the [[first fundamental form]] coefficient), so the integrand of the arc length integral can be written as <math>\sqrt{g_{ab} (u^a)' (u^b)'} </math> (where <math>u^1 = u </math> and <math>u^2 = v </math>).

=== Other coordinate systems ===
Let <math>\mathbf{C}(t) = (r(t), \theta(t))</math> be a curve expressed in polar coordinates.  The mapping that transforms from polar coordinates to rectangular coordinates is

:<math>\mathbf{x}(r,\theta) = (r\cos\theta, r\sin\theta ).</math>

The integrand of the arc length integral is  <math>|(\mathbf{x}\circ\mathbf{C})'(t)|.</math>  The chain rule for vector fields shows that <math>D(\mathbf{x}\circ \mathbf{C}) = \mathbf{x}_r r' + \mathbf{x}_{\theta} \theta'.</math>  So the squared integrand of the arc length integral is

:<math>(\mathbf{x_r}\cdot\mathbf{x}_r)(r')^2 + 2(\mathbf{x}_r\cdot\mathbf{x}_{\theta})r'\theta' + (\mathbf{x}_{\theta}\cdot\mathbf{x}_{\theta})(\theta')^2 = (r')^2 + r^2(\theta')^2.</math>

So for a curve expressed in polar coordinates, the arc length is

:<math>\int_{t_1}^{t_2} \sqrt{\left(\frac{dr}{dt}\right)^2 + r^2\left(\frac{d\theta}{dt}\right)^2 }dt = \int_{\theta(t_1)}^{\theta(t_2)} \sqrt{\left(\frac{dr}{d\theta}\right)^2 + r^2}d\theta.</math>

Now let <math>\mathbf{C}(t) = (r(t), \theta(t), \phi(t))</math> be a curve expressed in spherical coordinates where <math>\theta</math> is the polar angle measured from the positive <math>z</math>-axis and <math>\phi</math> is the azimuthal angle.  The mapping that transforms from spherical coordinates to rectangular coordinates is

:<math>\mathbf{x}(r,\theta,\phi) = (r\sin\theta\cos\phi, r\sin\theta\sin\phi, r\cos\theta).</math>

Using the chain rule again shows that <math>D(\mathbf{x}\circ\mathbf{C}) = \mathbf{x}_r r' + \mathbf{x}_{\theta}\theta' + \mathbf{x}_{\phi}\phi'.</math>  All dot products <math>\mathbf{x}_i \cdot \mathbf{x}_j</math> where <math>i</math> and <math>j</math> differ are zero, so the squared norm of this vector is

:<math>(\mathbf{x}_r\cdot \mathbf{x}_r )(r'^2) + (\mathbf{x}_{\theta} \cdot \mathbf{x}_{\theta})(\theta')^2 + (\mathbf{x}_{\phi}\cdot \mathbf{x}_{\phi})(\phi')^2 = (r')^2 + r^2(\theta')^2 + r^2 \sin^2\theta (\phi')^2.</math>

So for a curve expressed in spherical coordinates, the arc length is

:<math>\int_{t_1}^{t_2} \sqrt{\left(\frac{dr}{dt}\right)^2 + r^2\left(\frac{d\theta}{dt}\right)^2 + r^2\sin^2\theta \left(\frac{d\phi}{dt}\right)^2}dt. </math>

A very similar calculation shows that the arc length of a curve expressed in cylindrical coordinates is

:<math>\int_{t_1}^{t_2} \sqrt{\left(\frac{dr}{dt}\right)^2 + r^2\left(\frac{d\theta}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2}dt. </math>

== Simple cases ==

=== Arcs of circles ===
<!--
Arc length is not the same as [[arc measure]]. [What is that?] -->

Arc lengths are denoted by ''s'', since the Latin word for length (or size) is ''spatium''.

In the following lines, <math>r</math> represents the [[radius]] of a [[circle]], <math>d</math> is its [[diameter]], <math>C</math> is its [[circumference]], <math>s</math> is the length of an arc of the circle, and <math>\theta</math> is the angle which the arc subtends at the [[Centre (geometry)|centre]] of the circle. The distances <math>r, d, C,</math> and <math>s</math> are expressed in the same units.

* <math>C=2\pi r,</math> which is the same as <math>C=\pi d.</math>  This equation is a definition of [[Pi|<math>\pi.</math>]]
* If the arc is a [[semicircle]], then <math>s=\pi r.</math>
* {{anchor|Circular}}For an arbitrary circular arc:
** If <math>\theta</math> is in [[radian]]s then <math>s =r\theta.</math> This is a definition of the radian.
** If <math>\theta</math> is in [[Degree (angle)|degrees]], then <math>s=\frac{\pi r \theta}{180\ \text{deg}},</math> which is the same as <math>s=\frac{C \theta}{360\ \text{deg}}.</math>
** If <math>\theta</math> is in [[Grad (angle)|grads]] (100 grads, or grades, or gradians are one [[right-angle]]), then <math>s=\frac{\pi r \theta}{200\ \text{grad}},</math> which is the same as <math>s=\frac{C \theta}{400\ \text{grad}}.</math>
** If <math>\theta</math> is in [[Turn (geometry)|turns]] (one turn is a complete rotation, or 360°, or 400 grads, or <math>2\pi</math> radians), then <math>s=C \theta/\text{turn}</math>.

====Arcs of great circles on the Earth====
{{main|Great-circle distance}}
{{further|Geodesics on an ellipsoid}}

Two units of length, the [[nautical mile]] and the [[metre]] (or kilometre), were originally defined so the lengths of arcs of [[great circle]]s on the Earth's surface would be simply numerically related to the angles they subtend at its centre. The simple equation <math>s=\theta</math> applies in the following circumstances:

:* if <math>s</math> is in nautical miles, and <math>\theta</math> is in [[arcminute]]s ({{frac|60}} degree), or
:* if <math>s</math> is in kilometres, and <math>\theta</math> is in centigrades ({{frac|100}} [[Gradian|grad]]).

The lengths of the distance units were chosen to make the circumference of the Earth equal {{val|40000}} kilometres, or {{val|21600}} nautical miles. Those are the numbers of the corresponding angle units in one complete turn.

Those definitions of the metre and the nautical mile have been superseded by more precise ones, but the original definitions are still accurate enough for conceptual purposes and some calculations. For example, they imply that one kilometre is exactly 0.54 nautical miles. Using official modern definitions, one nautical mile is exactly 1.852 kilometres,<ref>{{cite web |url=http://www.physics.nist.gov/Pubs/SP811/appenB8.html |title=Special Publication 811 |first=Curt |last=Suplee |date=2 July 2009 |website=nist.gov}}</ref> which implies that 1 kilometre is about {{val|0.53995680}} nautical miles.<ref>[[CRC Handbook of Chemistry and Physics]], p. F-254</ref> This modern ratio differs from the one calculated from the original definitions by less than one part in 10,000.

=== Length of an arc of a parabola ===
{{for|a calculation of the length of a parabolic arc|Parabola#Arc length}}

==Historical methods==

===Antiquity===
For much of the [[history of mathematics]], even the greatest thinkers considered it impossible to compute the length of an irregular arc. Although [[Archimedes]] had pioneered a way of finding the area beneath a curve with his "[[method of exhaustion]]", few believed it was even possible for curves to have definite lengths, as do straight lines. The first ground was broken in this field, as it often has been in [[calculus]], by [[approximation]]. People began to inscribe [[polygon]]s within the curves and compute the length of the sides for a somewhat accurate measurement of the length. By using more segments, and by decreasing the length of each segment, they were able to obtain a more and more accurate approximation. In particular, by inscribing a polygon of many sides in a circle, they were able to find approximate values of [[pi (mathematical constant)|π]].<ref>{{Cite journal|last=Richeson|first=David|date=May 2015|title=Circular Reasoning: Who First Proved That C Divided by d Is a Constant?|journal=The College Mathematics Journal|volume=46|issue=3|pages=162–171|doi=10.4169/college.math.j.46.3.162|issn=0746-8342}}</ref><ref>{{Cite journal|last=Coolidge|first=J. L.|authorlink=Julian Coolidge|date=February 1953|title=The Lengths of Curves|journal=The American Mathematical Monthly|volume=60|issue=2|pages=89–93|doi=10.2307/2308256|jstor=2308256}}</ref>

===17th century===
In the 17th century, the method of exhaustion led to the rectification by geometrical methods of several [[transcendental curve]]s: the [[logarithmic spiral]] by [[Evangelista Torricelli]] in 1645 (some sources say [[John Wallis]] in the 1650s), the [[cycloid]] by [[Christopher Wren]] in 1658, and the [[catenary]] by [[Gottfried Leibniz]] in 1691.

In 1659, Wallis credited [[William Neile]]'s discovery of the first rectification of a nontrivial [[algebraic curve]], the [[semicubical parabola]].<ref>{{cite book|first=John|last=Wallis|title=Tractatus Duo. Prior, De Cycloide et de Corporibus inde Genitis…|location=Oxford|publisher=University Press|date=1659|url=http://gallica.bnf.fr/ark:/12148/bpt6k5759200j/f110.image|pages=91–96}}</ref> The accompanying figures appear on page 145. On page 91, William Neile is mentioned as ''Gulielmus Nelius''.

===Integral form===
Before the full formal development of calculus, the basis for the modern integral form for arc length was independently discovered by [[Hendrik van Heuraet]] and [[Pierre de Fermat]].

In 1659 van Heuraet published a construction showing that the problem of determining arc length could be transformed into the problem of determining the area under a curve (i.e., an integral).  As an example of his method, he determined the arc length of a semicubical parabola, which required finding the area under a [[parabola]].<ref>{{cite book|first=Hendrik|last=van Heuraet|url=https://books.google.com/books/ucm?id=lGFxGEEK52oC&pg=PA517#v=onepage&q&f=false|contribution=Epistola de transmutatione curvarum linearum in rectas [Letter on the transformation of curved lines into right ones]|title=Renati Des-Cartes Geometria|edition=2nd|location=Amsterdam|publisher=Louis & Daniel Elzevir|date=1659|pages=517–520}}</ref>  In 1660, Fermat published a more general theory containing the same result in his ''De linearum curvarum cum lineis rectis comparatione dissertatio geometrica'' (Geometric dissertation on curved lines in comparison with straight lines).<ref>{{cite book|author=M.P.E.A.S. (pseudonym of Fermat)|url=https://books.google.com/books?id=BBqoHZej2ZsC&pg=PA1#v=onepage&q&f=false|title=De Linearum Curvarum cum Lineis Rectis Comparatione Dissertatio Geometrica|location=Toulouse|publisher=Arnaud Colomer|date=1660}}</ref>

[[File:Arc length, Fermat.svg|thumb|300px|Fermat's method of determining arc length]]

Building on his previous work with tangents, Fermat used the curve

:<math> y = x^{3/2} \,</math>

whose [[tangent]] at ''x'' = ''a'' had a [[slope]] of

:<math> \textstyle {3 \over 2} a^{1/2} </math>

so the tangent line would have the equation

:<math> y = \textstyle {3 \over 2} {a^{1/2}}(x - a) + f(a). </math>

Next, he increased ''a'' by a small amount to ''a'' + ''ε'', making segment ''AC'' a relatively good approximation for the length of the curve from ''A'' to ''D''. To find the length of the segment ''AC'', he used the [[Pythagorean theorem]]:

: <math>\begin{align}
AC^2 &{}= AB^2 + BC^2 \\
&{} = \textstyle \varepsilon^2 + {9 \over 4} a \varepsilon^2 \\
&{}= \textstyle \varepsilon^2 \left (1 + {9 \over 4} a \right )
\end{align}</math>

which, when solved, yields

:<math>AC = \textstyle \varepsilon \sqrt { 1 + {9 \over 4} a\ }.</math>

In order to approximate the length, Fermat would sum up a sequence of short segments.

==Curves with infinite length{{anchor|Infinite}}==
{{see also|Coastline paradox}}
[[File:Koch curve.svg|thumb|The Koch curve.]] [[File:xsinoneoverx.svg|thumb|The graph of ''x''sin(1/''x'').]]
As mentioned above, some curves are non-rectifiable. That is, there is no upper bound on the lengths of polygonal approximations; the length can be made [[Mathematical jargon#arbitrarily large|arbitrarily large]]. Informally, such curves are said to have infinite length. There are continuous curves on which every arc (other than a single-point arc) has infinite length. An example of such a curve is the [[Koch snowflake|Koch curve]]. Another example of a curve with infinite length is the graph of the function defined by ''f''(''x'') =&nbsp;''x''&nbsp;sin(1/''x'') for any open set with 0 as one of its delimiters and ''f''(0) = 0. Sometimes the [[Hausdorff dimension]] and [[Hausdorff measure]] are used to quantify the size of such curves.

==Generalization to (pseudo-)Riemannian manifolds==
Let <math>M</math> be a [[pseudo-Riemannian manifold|(pseudo-)Riemannian manifold]], <math>\gamma:[0,1]\rightarrow M</math> a curve in <math>M</math> and <math>g</math> the (pseudo-) [[metric tensor]].

The length of <math>\gamma</math> is defined to be

:<math>\ell(\gamma)=\int_{0}^{1} \sqrt{ \pm g(\gamma'(t),\gamma '(t)) } \, dt, </math>

where <math>\gamma'(t)\in T_{\gamma(t)}M</math> is the tangent vector of <math>\gamma</math> at <math>t.</math>  The sign in the square root is chosen once for a given curve, to ensure that the square root is a real number. The positive sign is chosen for spacelike curves; in a pseudo-Riemannian manifold, the negative sign may be chosen for timelike curves. Thus the length of a curve is a non-negative real number. Usually no curves are considered which are partly spacelike and partly timelike.

In [[theory of relativity]], arc length of timelike curves ([[world line]]s) is the [[proper time]] elapsed along the world line, and arc length of a spacelike curve the [[proper distance]] along the curve.

==See also==
* [[Arc (geometry)]]
* [[Circumference]]
* [[Crofton formula]]
* [[Elliptic integral]]
* [[Geodesic]]s
* [[Intrinsic equation]]
* [[Numerical integration|Integral approximations]]
* [[Line integral]]
* [[Meridian arc]]
* [[Multivariable calculus]]
* [[Sinuosity]]

==References==
{{reflist}}

==Sources==
* {{cite book|last=Farouki|first=Rida T.|date=1999|contribution=Curves from motion, motion from curves|editor1-first=P.-J.|editor1-last=Laurent|editor2-first=P.|editor2-last=Sablonniere|editor3-first=L. L.|editor3-last=Schumaker|title=Curve and Surface Design: Saint-Malo 1999|pages=63–90|publisher=Vanderbilt Univ. Press|isbn=978-0-8265-1356-4}}

==External links==
* {{springer|title=Rectifiable curve|id=p/r080130}}
<!-- * [http://math.kennesaw.edu/~jdoto/13.pdf Math Before Calculus] not working -->
* [https://web.archive.org/web/20071106083431/http://www3.villanova.edu/maple/misc/history_of_curvature/k.htm The History of Curvature]
* {{MathWorld|title=Arc Length|urlname=ArcLength}}
* [http://demonstrations.wolfram.com/ArcLength/ Arc Length] by [[Ed Pegg, Jr.]], [[The Wolfram Demonstrations Project]], 2007.
* [http://www.pinkmonkey.com/studyguides/subjects/calc/chap8/c0808501.asp{{dead link|date=July 2017 |bot=InternetArchiveBot |fix-attempted=yes }} Calculus Study Guide – Arc Length (Rectification)]
* [http://www-groups.dcs.st-and.ac.uk/~history/Curves/Curves.html Famous Curves Index] ''The MacTutor History of Mathematics archive''
* [http://demonstrations.wolfram.com/ArcLengthApproximation/ Arc Length Approximation] by Chad Pierson, Josh Fritz, and Angela Sharp, [[The Wolfram Demonstrations Project]].
* [https://web.archive.org/web/20110720095511/http://numericalmethods.eng.usf.edu/experiments/Length_of_curve_experiment.pdf Length of a Curve Experiment] Illustrates numerical solution of finding length of a curve.

{{DEFAULTSORT:Arc Length}}
[[Category:Integral calculus]]
[[Category:Curves]]
[[Category:Length]]