# Talk:Arc length

WikiProject Mathematics (Rated B-class, High-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 B Class
 High Importance
Field:  Geometry

I think it should be redirected to curve

Tosha 01:25, 15 Mar 2004 (UTC)

I completely disagree with you, Tosha. I think this deserves a page on its own.
This means that in discussion of the geometry of the curves, any links to arc length now jump directly to the curves article. This is a disservice to readers navigating these articles who expect to go directly to a discussion of the concept of arc length, rather than to to the article on curves. -- Decumanus 19:39, 15 Mar 2004 (UTC)
I agree; this page can stand on its own. Michael Hardy 21:34, 15 Mar 2004 (UTC)

Ok, I think this way it is harder to keep it uptodate, but do as you wish.Tosha 23:56, 15 Mar 2004 (UTC)

This article is incomplete. It needs cleanup. I wikified the maths, but some of the formulas didn't seem to make sense the way they were written, so they should be checked.

The article is also in sore need of some images.

I reordered the article and fixed some of the math. It still needs more work though -Jacob 01:38, 14 June 2005 (UTC)

I've been fixated on the text bit - now that I look closely, that explanation of the Fermat math doesn't look right. It keeps sprouting unexplained new variables. Tearlach 19:37, 16 July 2005 (UTC)
I only fixed the LaTex formating of the math -not the content- now that I look criticaly, I see what you mean. It looks like the original author of the page just rewrote it from the pdf he cites. Reading the file explains what he was getting at. We I get some more time, I'll make a plot so one can follow along. We should add the final part too (from the pdf) where the formula shows that Fermat really did find the integral and his formula, he just didn't understand the implications. I'd like another source for verification that this is how it historicaly happened.
As far as I understand it, both van Heuraet and Fermat derived that if you have a function f(x), then the arc length is given by the area under a new function g(x) = sqrt(1 + f'(x)^2). So they had the function to be integrated, but not the integral calculus to apply to it. Tearlach 11:49, 17 July 2005 (UTC)

A bit added about the problems. As it stood, it was overoptimistic about the benefits of applying calculus. PS Semicubical parabola is y^3 = a*x^2. Tearlach 23:24, 15 July 2005 (UTC)

More bibliographic detail: Heuraet at MacTutor, Rida T Farouki reference here (PostScript) and here (text). Tearlach 07:26, 16 July 2005 (UTC)

And this Wallis biography from WW Rouse Ball describes van Heuraet's method for curve rectification. Tearlach 13:07, 16 July 2005 (UTC)

I added a graph for Fermat's method. I had different colors for the line originaly, but somehow, gnuplot stoped changing them as the code evolved. See Image:Arc length, Fermat.png for the code and please help me fix it. --Jacob 18:09, 22 July 2005 (UTC)

The location "Arc length" is arrived at from "Curve length". No idea how history brought us to this situation. Clearly, this article should be "Curve length", and "Arc length" should redirect to "Arc_(geometry)". 68.84.233.37 (talk) 10:50, 30 June 2013 (UTC)

## Article title

According to the article Arc, an arc is a continuous portion of a circle; part of a circle's circumference (also called a circle segment). The present article is about curves. It uses the undefined term "irregular arc", but in fact even in the present text the use of the term "curve" dominates the use of the term "arc". My conclusion is that the title ought to be: Length of a curve. LambiamTalk 03:09, 9 April 2006 (UTC)

Actually, I prefer arc length, as that is what the concept is usually called in calculus books and more advanced books on differential geometry. The other names should just redirect to arc length. I also suspect that when writing an article, most mathematicians would use arc length, so this avoids unnecessary redirects.
There are terminology issues, as you've noticed. "Arc" actually is used to refer to more than what is currently in arc. In geometry/topology areas, it actually means a continuous map of the unit interval, and often in differential geometry, "arc" refers to "rectifiable arc". This usage of "arc" is actually fairly old, and that is the "arc" that is referred to by "arc length"; it may be tied to the historical "irregular arc" mentioned in the article. I think, though, that more people prefer "path" or "curve" over "arc" nowadays in topology.
In answer to your objection, while it may be strange that "curve" dominates "arc" in the article, it's just an artifact of history and usage that "arc length" has become the standard term rather than "curve length", although of course, people would probably understand what you mean by the latter in the appropriate context. --C S (Talk) 03:58, 9 April 2006 (UTC)
I did a bit of research on other treatments. Mathworld, under "arc length", defines "Arc length is defined as the length along a curve"[1] and proceeds to give a treatment that will blow away all non-mathematicians; it appears to presuppose differentiability. PlanetMath, under "arc length", defines: "Arclength is the length of a section of a differentiable curve"[2] (italics added, L.), and the treatment given corresponds more or less to that of our Length of an arc. We also have Curve#Lengths_of_curves, which is somewhat unreadable and probably quite incomprehensible to the mathematically untrained, but in essence what I believe the Length article should cover: length as a kind of limit, with differentiable curves in Euclidean 2-space as a special case. Alternatively, we should make clear that this article covers only an important but nevertheless special case and insert a link to the more general, unfortunately unreadable, treatment. LambiamTalk 06:19, 9 April 2006 (UTC)

I prefer arc length, I'd like to see this article moved to there. -lethe talk + 17:43, 9 April 2006 (UTC)

I also prefer arc length. The more general stuff at curve I think is also called arc length, but in the case of metric spaces, so it should be treated here. --MarSch 10:24, 11 April 2006 (UTC)
I have moved the article. This required deleting and merging old history, since there was once an article arc length, which has a history, and was improperly made into a redirect by Tosha in 2004, in violation of the GFDL. -lethe talk + 19:42, 11 April 2006 (UTC)

## Length of Circular Arc Segments

As this article is named Arc Length, I believe it should at its outset completely and clearly address the generally understood notion of Arc pertaining to a segment of a circle before it addresses the arc lengths of curves. The current definition of a circular arc, while technically correct is woefully inadequate to general users and difficult to find on the page. In order to make Wikipedia useful to the general populace, think like and write to non-mathematicians first, then delve into the higher notions of the concept.

I completely agree with this. This is an endemic issue in most of the math sections on Wikipedia. Usually little to no regard is given to those who are seeking to understand the basics of the concepts and the articles dive headlong into the higher math without giving adequate explanations of what the formulas mean. Most people who are looking for information on Wikipedia don't "speak math" like we do and often skip right over the formulas. I know that the formulas are the most concise definition, but only for those that already understand the concepts or know how to speak mathematics. In the effort to be concise and complete the usefulness of the encyclopedic information often becomes nullified. It is easy to forget who our customers are, who uses the information and what is its purpose? Start with the most basic commonly held definitions give them a solid foundation and build on them. Dissymmetry (talk) 14:46, 7 June 2012 (UTC)

## standard definition

The standard definition of arc length---the sup of polygonal path lengths---is not even mentioned here. I'll be back..... Michael Hardy 22:51, 7 May 2006 (UTC)

That definition applies to all curves in a metric space, not just rectifiable arc segments (differentiable curves?) in Euclidean space, in which case one might wonder why the article is not called "Curve length" or something like that. See also the above discussion on the article title. If someone feels like improving the present article, then note that it never defines what an "irregular arc segment" is, does not give any definition of arc length but only ways to determine it, and further confuses functions with their graphs plotted as curves in space. I also find the account under "Ancient" rather dubious and doubt that it is true as stated.
If general curves are treated, it would be reasonable to give fractal curves as an example of curves of infinite length; see also How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. --LambiamTalk 23:25, 7 May 2006 (UTC)

"Rectifiable" does not imply "differentiable". I thought "rectifiable" meant the total length is finite, regardless of differentiability, or at least the length between any two points on the curve is finite (as with a line). Michael Hardy 03:13, 9 May 2006 (UTC)

The parenthetical question was a guess at a possibly intended meaning of "arc segment", given PlanetMath's definition of "arc length" as "the length of a section of a differentiable curve". --LambiamTalk 08:44, 9 May 2006 (UTC)

In regard to curves that have infinite length between two points on the curve, one could see also space-filling curve. Michael Hardy 21:35, 9 May 2006 (UTC)

Indeed, a special but well-known case of fractals having Hausdorff dimension equal to 2.

Isn't there also a definition based on the limits of a cover of balls? --njh 11:51, 16 May 2006 (UTC)

Both Hausdorff dimension and Minkowski-Bouligand dimension say something like: I am what you get as a limit of a cover of balls. Both are written in a way that makes it hard to see how this relates to the more formal definitions given. How they can be "informally" the same but actually different is not immediately clear to me. --LambiamTalk 13:12, 16 May 2006 (UTC)
They talk about computing the dimension of a structure. I recall there being a way to define arc length itself as a covering. --njh 02:19, 17 May 2006 (UTC)

## Derivation

A representative linear element of the function ${\displaystyle {\begin{cases}y=t^{5}\\x=t^{3}\end{cases}}}$

I would like to show the derivation of at least the fundamental parametric arc length formula, however since I'm new to this math format and it is a moderately long explination, I'm going to be working on it here and moving it into the article once it is complete and has been peer reviewed. Please let me know of any errors that you may see, I will move it into the main page in the near future. If anyone might be able to explain the conversion of this integral into the one used for ordinary functions (f(x)) it would be very useful.

In order to approximate the arc length of the curve, it is split into many linear segments. To make the value exact, and not an approximation, we will need infinitely many linear elements. This means that each element is infinately small. This fact manifests itself later on when we use an integral.

We start by looking at a representative linear segment (see image) and observe that its length (element of the arc length) will be the differential ds. We will call the horizontal element of this distance dx, and the vertical element dy.

Now, recall the distance formula which tells us that

${\displaystyle ds={\sqrt {dx^{2}+dy^{2}}}}$.

Since the function is defined in time, we add up the segments (ds) across infintessimally small intervals of time (dt) yielding the integral:

${\displaystyle \int _{a}^{b}{\sqrt {{\bigg (}{\frac {dx}{dt}}{\bigg )}^{2}+{\bigg (}{\frac {dy}{dt}}{\bigg )}^{2}}}\,dt}$

Which is the arc length from ${\displaystyle t=a}$ to ${\displaystyle t=b}$ of the parametric function f(t).

For example, the curve in this figure if defined by:

${\displaystyle {\begin{cases}y=t^{5}\\x=t^{3}\end{cases}}}$

Subsequently, the arc length integral fo values of t from −1 to 1 is:

${\displaystyle \int _{-1}^{1}{\sqrt {(3t^{2})^{2}+(5t^{4})^{2}}}\,dt=\int _{-1}^{1}{\sqrt {9t^{4}+25t^{8}}}\,dt}$

Using computational approximations, we can obtain a very accurate (but still approximate) arc length of 2.905.

48v 03:18, 12 July 2006 (UTC)

This has now been added to the article. 48v 21:30, 14 July 2006 (UTC)

A revision of the first image, swapping labels dx and dy is at http://en.wikipedia.org/wiki/File:Arc_length_approximation.svg. It also has the ds label moved. This article's preceding section says, "Consider an infinitesimal part of the curve ds (or consider this as a limit in which the change in s approaches ds)." which seems to allow either the portion of curve within the triangle or the straight line which approximates it to be labeled ds. I'm not sure which is more proper. Should the revised images have the ds label at the original location?

96.48.157.217 (talk) 13:59, 16 April 2013 (UTC)

Whbny (talk) 21:06, 4 February 2018 (UTC)whbny %%% The last derivation regarding arc length being independent of parametrization has a few serious problems. First, a continuously differentiable bijection does not need to be non-decreasing: for example, \phi(x)=-x. Second, the inverse of a continuously differentiable bijection need not be differentiable: consider the bijection \phi(x)=x^3, whose inverse bijection fails to be differentiable at x=0. In fact, the derivative of \phi^{-1}(x) becomes unbounded around x=0. These matters can be fixed (I can fix them but I need to get more accustomed to using this Talk page and format).

## Notations used in formulae

The lowercase 'e' in italics is used for Euler's Number. The symbol used in this context should be 'ε' or 'є,' not e.

The Roc 1217 22:28, 4 April 2007 (UTC)

## Derivation without parametric

I think the article is missing a good demonstration not involving the parametric form. This is my work on it, and I think it could be included in the article, though a more rigorous treatment is welcome.

We have a function f(x). It is continuous in the interval [a,b]. Now, let us divide this interval in equal parts, Δx. Starting from a, each successive increment in x, will also increment the f(x), and the arc lenght of that part will be given as:

${\displaystyle L_{i}={\sqrt {\Delta x^{2}+[f(a+i\Delta x))-f(a+(i-1)\Delta x]^{2}}}}$

Foe example, the first increment will result in an arc length of approximately:

${\displaystyle L_{1}={\sqrt {\Delta x^{2}+[f(a+\Delta x))-f(a)]^{2}}}}$

The total arc length will be:

${\displaystyle L_{T}=\sum _{a}^{b}{\sqrt {\Delta x^{2}+[f(a+i\Delta x))-f(a+(i-1)\Delta x]^{2}}}}$
${\displaystyle L_{T}=\sum _{a}^{b}{\sqrt {\Delta x^{2}\left(1+\left[{\frac {f(a+i\Delta x)-f(a+(i-1)\delta x]}{\Delta x}}\right]^{2}\right)}}}$
${\displaystyle L_{T}=\sum _{a}^{b}\Delta x{\sqrt {1+\left[{\frac {f(a+i\Delta x)-f(a+(i-1)\delta x]}{\Delta x}}\right]^{2}}}}$

Now, taking the limit as Δx becomes infinitesimal, and summing up all the parts, we get the expression:

${\displaystyle L_{T}=\int _{a}^{b}{\sqrt {1+[f'(x)]^{2}}}\,dx}$

Goldencako 00:41, 7 November 2007 (UTC)

## First section, "General approach", needs work

"A curve in, say, the plane can be approximated by connecting a finite number of points on the curve using line segments to create a polygonal path. Since it is straightforward to calculate the length of each linear segment using the theorem of Pythagoras, the total length of the approximation can be found by summing the lengths of each linear segment.

Using a larger number of segments (with each segment often of a smaller length) usually provides a better approximation to the curve, and the length of such an approximation will be greater than one using fewer segments.

For some curves it is the case that none of the lengths of the approximations exceeds a certain smallest number L. If such a number exists, then the curve is said to be rectifiable and the curve is defined to have arc length L."

The first paragraph should be removed entirely, since above all, how to calculate arclength for practical purposes is a separate issue from how arclength is defined -- the definition is more basic and should come first and be clearly labeled as such. (Even if it were appropriate to begin with a discussion of how to *calculate* arclength, this paragraph makes no statement that the cartesian coordinates of the segments' endpoints are known -- which makes the mention of "using the theorem of Pythagoras" irrelevant.)

The word "usually" in the second paragraph seems strange. To make sense, it needs to be stated what a "better approximation to the curve" means, if this phrase is to be used.

The third paragraph is false as stated. (E.g., the length of each polygonal approximation to a unit circle does not exceed 99, but 99 exceeds 2π, the arclength of this circle.)Daqu (talk) 09:08, 17 August 2008 (UTC)

In order to create this new section I separated out statements about the intuition behind calculating arc length, and provided a better definition (see the history for more details), so some of it is a re-write of stuff previously included. The previous version had intuitive statements about arc length being defined for R2, and I thought that keeping these ideas in the context of the plane was a good way of introducing the idea, since, perhaps, most people might find the idea of a curve in n-dimensional space a little intimidating.
I disagree that the definition is more basic than the intuitive version, for two reasons: (1) It's not more basic for non-mathematicians; (2) The definition is precisely that intuitive concept given symbolic form.
The language for this section is written informally: if you think it needs work, go ahead and make the changes. And if it's imprecise, go ahead and make those changes too. It's anybody's article.
Finally, the cartesian coordiates of the segments' endpoints is relevant only if one is working in Euclidean space; if X were a general metric space, I don't know what "cartesian coordinates" would mean. Xantharius (talk) 21:37, 17 August 2008 (UTC)

## Slightly different way should be removed

This section is flagged with a cleanup, but it's not worth cleaning up. It's not substantially different from the main derivation, and full of grammatical and notational errors. I suggest it be removed. Bryanclair (talk) 20:29, 31 March 2009 (UTC)

I second this removal.—Tetracube (talk) 20:35, 31 March 2009 (UTC)

## Time?

Would it not be better for the derivation to call t a parameter rather than saying "time"? 149.157.1.184 (talk) 16:53, 16 May 2009 (UTC)

## Incorrect use of term "Straight Line"

Within section "Historical methods", subsection "Ancient", a 'straight line' is said to have a definite length. According to MathWorld a line (sometimes called a 'straight line') is "a straight one-dimensional figure having no thickness and extending infinitely in both directions".

In the context of this article, I believe the correct term should be 'line segment' which does have a definite length. Fhv1374 (talk) 08:02, 3 December 2009 (UTC)

## petal

Hi. What means petal here ? petal of Lea-Fatou flower  ? --Adam majewski (talk) 18:08, 18 February 2013 (UTC)

## incorrect edit by prodigy

The recent change to the formula for arclength is in error. Tkuvho (talk) 09:12, 29 December 2014 (UTC)

## General approach section

I removed the following paragraph from this section:

Polygonal approximations are linearly dependent on the curve in a few select cases. One of these cases is when the curve is simply a point function as is its polygonal approximation. Another case where the polygonal approximation is linearly dependent on the curve is when the curve is linear. This would mean the approximation is also linear and the curve and its approximation overlap. Both of these two circumstances result in an eigenvalue equal to one. There are also a set of circumstances where the polygonal approximation is still linearly dependent but the eigenvalue is equal to zero. This case is a function with petals where all points for the polygonal approximation are at the origin.

"eigenvalue" and "linearly dependent" make no sense here. The entire paragraph is confusing, and the point it seems to be trying to make is inappropriate in a section that's trying to give a gentle introduction by explaining the basic idea. — Preceding unsigned comment added by 24.56.116.202 (talk) 17:00, 10 August 2015 (UTC)

## Definition

Tosha, do you mean the curve should be defined as a piecewise differentiable mapping from an interval to ${\displaystyle \mathbb {R} ^{n}?}$ This would certainly be a more useful definition for the vast majority of applications. Jrheller1 (talk) 23:38, 15 February 2016 (UTC)

## Example of numerical integration

This information would be useful to anybody who writes a program that needs to calculate the arc length of a curve. It shows that the fact that there is usually no closed form solution for the arc length integral doesn't really matter because numerical integration (even very accurate numerical integration) is usually very efficient. Gauss-Kronrod rules and Gaussian quadrature rules are just the standard ways of doing numerical integration of a non-singular integrand. Jrheller1 (talk) 05:13, 5 March 2016 (UTC)

For theoretical purposes, especially on a general curve of a particular family (such as an arbitrary parabola for which you don't know its parameters), a closed-form solution is much preferable, even if given in terms of special functions like elliptic integrals. Anyone writing a program to compute arc length would probably use numerical integration even if the curve possessed a closed-form arc length formula, since even if it exists, it may be unsuitable for numerical purposes. More generally, the efficiency of numerical integration does no good for theoretical purposes for any definite integral in general. As someone focusing on pure math (at least for now), it only distracts me when I see a remark on numerical integration.--Jasper Deng (talk) 07:59, 5 March 2016 (UTC)
I find the practical information useful; the article needs not be just theoretical. fgnievinski (talk) 00:51, 12 March 2016 (UTC)
If someone wants to do numerical integration they can head over to the article on that. Why is adding numerical integration here any more useful than other situations not determinable in closed form, such as elliptic integrals and the error function?--Jasper Deng (talk) 21:19, 15 March 2016 (UTC)
Numerical integration is applied to many many different mathematical problems; it is impractical for all of them to be covered in a single article. As the OP explained, there are some nuances to this application, that are not obvious to the average reader. The error function can be consider a closed-form_expression#Alternative definitions. fgnievinski (talk) 00:59, 16 March 2016 (UTC)
But how are any of these nuances specific to the problem of arc length? The OP only mentions that these are generally applicable numerical integration methods, without discussion of how arc length integrands are any more suited to those methods than general integrands.--Jasper Deng (talk) 08:25, 16 March 2016 (UTC)
I'd support summarizing that sub-section as a single sentence, just mentioning that quadrature can be readily applied after finding a suitable parameterization; you're right in that currently not much is said about its details (which can be quite complicated); the lengthy numerical example is not particularly illuminating. fgnievinski (talk) 09:42, 16 March 2016 (UTC)
I think that would be satisfiable for me too. How about ?--Jasper Deng (talk) 07:19, 17 March 2016 (UTC)

The history of the page shows that there was an example of numerical integration for more than 9 years and nobody had a problem with it all that time. While removing several redundant and not very good derivations of the arc length integral, I removed this example temporarily back in February 2016. Then I replaced the previous example with a better example. In this new example, the integral has an easily computed transcendental value, so it is easier to analyze the error of Gauss or Gauss-Kronrod quadrature (or any other numerical integration method). How could anybody possibly have a problem with this improved example? Jrheller1 (talk) 04:16, 28 April 2016 (UTC)

That's not relevant. My issue is with including an example at all.--Jasper Deng (talk) 15:23, 28 April 2016 (UTC)

## Smooth or C1

In the section Arc length#Definition for a smooth curve it starts by saying

Let ${\displaystyle f:[a,b]\to \mathbb {R} ^{n}}$ be a continuously differentiable function.

which means that the function is ${\displaystyle C^{1}}$, the remainder of the section seems to work for ${\displaystyle C^{1}}$ curves. The definition of a "smooth" function is that it is ${\displaystyle C^{\infty }}$. Should the section be renamed "Definition for a differentiable curve", or am I missing something?--Salix alba (talk): 05:10, 10 May 2016 (UTC)

"Once continuously differentiable" or ${\displaystyle C^{1}}$ is more precise than "smooth". Sometimes ${\displaystyle C^{1}}$ or ${\displaystyle C^{2}}$ curves or surfaces are referred to as "smooth", though. A ${\displaystyle C^{1}}$ or ${\displaystyle C^{2}}$ curve appears to be "smooth" (using the word smooth in a physical sense rather than strict mathematical sense). Jrheller1 (talk) 02:00, 13 May 2016 (UTC)
I'm agnostic whether narrowing the discussion to ${\displaystyle C^{1}}$ is important (Wikipedia generally eschews highly technical arguments, and "smooth" is often understood by context), but I object to the use of ${\displaystyle |f'(t)|}$ rather than ${\displaystyle ||f'(t)||}$. The function ${\displaystyle f(t)}$ and its derivative are vector valued, so the (Euclidean) norm rather than absolute value is needed in the arc length formula.Hardmath (talk) 06:12, 27 December 2017 (UTC)
Single vertical bars are often used for vector norm, for example in Rudin (one of the sources cited). Jrheller1 (talk) 16:40, 28 December 2017 (UTC)

## Unnecessary verbiage

The words "where the supremum is taken over all possible partitions ${\displaystyle a=t_{0} of ${\displaystyle [a,b]}$" are perfectly clear. But someone keeps inserting the unnecessary and distracting verbiage "please note ${\displaystyle N}$ is not a fixed number, i.e., the supemum is taken over all possible finite partitions of the interval ${\displaystyle [a,b]}$, where the curve ${\displaystyle f}$ is defined." Jrheller1 (talk) 04:52, 27 September 2016 (UTC)

Hello fellow Wikipedians,

I have just modified one external link on Arc length. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

When you have finished reviewing my changes, please set the checked parameter below to true or failed to let others know (documentation at {{Sourcecheck}}).

An editor has reviewed this edit and fixed any errors that were found.

• If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
• If you found an error with any archives or the URLs themselves, you can fix them with this tool.

Cheers.—InternetArchiveBot 03:26, 17 October 2016 (UTC)

Links checked and found to be ok, although this list might need some trimming. - DVdm (talk) 09:05, 17 October 2016 (UTC)