Talk:Arc length

message: In the second section the mean value theorem is used on a vector-valued function, but the theorem only holds for real-valued functions. For example the proof doesn't hold for the curve f(t)=(\cos(t),\sin(t),t) for t\in [0,2\pi], since f'(t)=(-\sin(t),\cos(t),1).
An alternative proof of the arc length formula could rely on Fatou's lemma. If \text{len}(f) is at first defined using the integral formula one can show that \text{len}(f)=\text{sup}_{\text{partitions of [a,b]}}\sum_i |f(t_{i+1})-f(t_i)| by (i) showing by direct inequality that a straight line is the shortest path between two points (giving \geq) (ii) \text{len} is lower semi-continuous according to pointwise convergence of the derivatives of curves (using Fatou's lemma) (iii) every smooth path (C^1) has a succession of piecewise linear curves with vertices on the curve whose derivatives converge pointwise to the derivative of the path (giving \leq).
One could also prove the result for when n=2  using only the mean value theorem and the implicit function theorem.  — Preceding unsigned comment added by 87.6.200.188 (talk) 17:07, 26 April 2020 (UTC)
thread_title: Error in proof

message: 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)
thread_title:  Length of Circular Arc Segments 

message: 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 
: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:
:\int_a^b \sqrt{\bigg(\frac{dx}{dt}\bigg)^2+\bigg(\frac{dy}{dt}\bigg)^2}\,dt
Which is the arc length from t=a to t=b of the parametric function f(t).
For example, the curve in this figure if defined by:
:\begin{cases} y = t^5 \\ x = t^3 \end{cases}
Subsequently, the arc length integral fo values of t from −1 to 1 is:
:\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)
replies: text: This has now been added to the article. 48v 21:30, 14 July 2006 (UTC)
thread_title: Derivation

message: 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)
thread_title:  Notations used in formulae 

message: 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:
: 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:
:L_1 = \sqrt{\Delta x^2 + [f(a+\Delta x))-f(a)]^2}
The total arc length will be:
:L_T = \sum_{a}^{b} \sqrt{\Delta x^2 + [f(a+i\Delta x))-f(a+(i-1)\Delta x]^2}
: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)}
: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:
:L_T = \int_{a}^{b}\sqrt{1 + [f'(x)]^2}\, dx
Goldencako 00:41, 7 November 2007 (UTC)
thread_title:  Derivation without parametric 

message: 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)
replies: text: I second this removal.—Tetracube (talk) 20:35, 31 March 2009 (UTC)
thread_title: Slightly different way should be removed

message: 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)
thread_title:  Time? 

message: 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)
thread_title:  Incorrect use of term "Straight Line" 

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

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

message: 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)
thread_title:  General approach section 

message: Tosha, do you mean the curve should be defined as a piecewise differentiable mapping from an interval to \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)
thread_title:  Definition 

message: 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)
replies: text: 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)
replies: text: I find the practical information useful; the article needs not be just theoretical. fgnievinski (talk) 00:51, 12 March 2016 (UTC)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text: I think that would be satisfiable for me too. How about ?--Jasper Deng (talk) 07:19, 17 March 2016 (UTC)

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

message: In the section Arc length#Definition for a smooth curve it starts by saying 
:Let f:[a,b]\to\R^n be a continuously differentiable function.
which means that the function is C^1, the remainder of the section seems to work for C^1 curves. The definition of a "smooth" function is that it is 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)
replies: text: "Once continuously differentiable" or C^1 is more precise than "smooth".  Sometimes C^1 or C^2 curves or surfaces are referred to as "smooth", though. A C^1 or 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)

text: I'm agnostic whether narrowing the discussion to C^1 is important (Wikipedia generally eschews highly technical arguments, and "smooth" is often understood by context), but I object to the use of |f'(t)| rather than ||f'(t)||.  The function 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)
replies: text:  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)
thread_title:  Smooth or C1 

message: The words "where the supremum is taken over all possible partitions a=t_0<t_1<\dots <t_{N-1}<t_N=b of [a,b]" are perfectly clear.  But someone keeps inserting the unnecessary and distracting verbiage "please note N is not a fixed number, i.e., the supemum is taken over all possible finite partitions of the interval [a,b], where the curve f is defined."  Jrheller1 (talk) 04:52, 27 September 2016 (UTC)
replies: text: The words "for any N" should suffice. 2601:200:C000:1A0:6923:FA76:F1E6:D668 (talk) 17:17, 16 July 2022 (UTC)
thread_title:  Unnecessary verbiage 

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

message: The section Formula for a smooth curve begins as follows:
"Let f\colon[a,b]\to\R^n be an injective and continuously differentiable (i.e., the derivative is a continuous function) function. The length of the curve defined by f can be defined as the limit of the sum of linear segment lengths for a regular partition of [a,b] as the number of segments approaches infinity. This means
L(f) = \lim_{N\to\infty}\sum_{i=1}^N \bigg|f(t_i) - f(t_{i-1})\bigg|
where t_i = a + i(b - a)/N = a + i\Delta t with \Delta t = \frac{b-a}{N} = t_i - t_{i-1} for i = 0, 1, \dotsc, N. This definition is equivalent to the standard definition of arc length as an integral:
L(f) = \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."
But the appropriate notation for the distance between points in n-dimensional space is not the absolute value sign: | |, but instead it is the symbol for a norm:  ⃦  ⃦. 2601:200:C000:1A0:6923:FA76:F1E6:D668 (talk) 17:15, 16 July 2022 (UTC)
replies: text: Absolutely right! The argumentation is really misleading because it coincides with total variation. 149.139.33.244 (talk) 10:17, 21 July 2022 (UTC)
thread_title:  Norm, not absolute value 

message: "Arc length is the distance between two points along a section of a curve."
Isn't the distance between two points the length of the shortest segment that has those two points as endpoints? This is also what the Wikipedia page on distance says (if I interpreted it correctly). Wouldn't it be more appropriate to define arc length as the length of an arbitrary curve with two endpoints? Or if one doesn't wish to reuse the word length, the distance traveled along a curved path from a point to another point? I don't like this last definition because it kind of implies that something has traveled along the arc but anyways I don't mean to come up with a definition yet, I would just like to know if the definition on the Wikipedia page for arc length is correct. 131.114.35.40 (talk) 14:06, 26 October 2022 (UTC)
replies: text: I agree, a better introductory definition would be: "The arc length is the length of a rectifiable curve." Stqckfish (talk) 17:37, 28 March 2024 (UTC)
thread_title:  Definition 

message: In my opinion, the introduction of this article on the definition of a rectifiable curve is very unclear and it repeats itself, making it confusing.
Therefore, I would propose adding a section entitled "Formal definitions." In this section, I would present the following definitions, and show that they are, indeed, equivalent:
1. For any sequence of partitions of [a,b] such that the lengths of the corresponding line segments of the curve converge to 0, the sequence of the sums of these lengths converges to the value L, the arc length.
2. The supremum of the set of all sums of lengths corresponding to a partition on [a,b] exists.
The first statement is the intuitive and mathematically rigorous definition of arc length.
These definitions could be worded better and made more understandable with mathematical notation. I believe this would clear up confusion and make the whole article more accurate. 
 Stqckfish (talk) 17:29, 28 March 2024 (UTC)
thread_title:  Adding a definition section