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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 for , since .
An alternative proof of the arc length formula could rely on Fatou's lemma. If is at first defined using the integral formula one can show that by (i) showing by direct inequality that a straight line is the shortest path between two points (giving ) (ii) is lower semi-continuous according to pointwise convergence of the derivatives of curves (using Fatou's lemma) (iii) every smooth path () has a succession of piecewise linear curves with vertices on the curve whose derivatives converge pointwise to the derivative of the path (giving ).
One could also prove the result for when 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)
I think it should be redirected to curve
Tosha 01:25, 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)
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)
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. Lambiam Talk 03:09, 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)
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.
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)
"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)
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)
Isn't there also a definition based on the limits of a cover of balls? -- njh 11:51, 16 May 2006 (UTC)
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
Since the function is defined in time, we add up the segments (ds) across infintessimally small intervals of time (dt) yielding the integral:
Which is the arc length from to of the parametric function f(t).
For example, the curve in this figure if defined by:
Subsequently, the arc length integral fo values of t from −1 to 1 is:
Using computational approximations, we can obtain a very accurate (but still approximate) arc length of 2.905.
48v 03:18, 12 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).
The lowercase 'e' in italics is used for Euler's Number. The symbol used in this context should be 'ε' or 'є,' not e.
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:
Foe example, the first increment will result in an arc length of approximately:
The total arc length will be:
Now, taking the limit as Δx becomes infinitesimal, and summing up all the parts, we get the expression:
Goldencako 00:41, 7 November 2007 (UTC)
This section reads as follows:
"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)
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)
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)
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)
Hi. What means petal here ? petal of Lea-Fatou flower ? -- Adam majewski ( talk) 18:08, 18 February 2013 (UTC)
The recent change to the formula for arclength is in error. Tkuvho ( talk) 09:12, 29 December 2014 (UTC)
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)
Tosha, do you mean the curve should be defined as a piecewise differentiable mapping from an interval to This would certainly be a more useful definition for the vast majority of applications. Jrheller1 ( talk) 23:38, 15 February 2016 (UTC)
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)
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)
In the section Arc length#Definition for a smooth curve it starts by saying
which means that the function is , the remainder of the section seems to work for curves. The definition of a "smooth" function is that it is . Should the section be renamed "Definition for a differentiable curve", or am I missing something?-- Salix alba ( talk): 05:10, 10 May 2016 (UTC)
The words "where the supremum is taken over all possible partitions of " are perfectly clear. But someone keeps inserting the unnecessary and distracting verbiage "please note is not a fixed number, i.e., the supemum is taken over all possible finite partitions of the interval , where the curve is defined." Jrheller1 ( talk) 04:52, 27 September 2016 (UTC)
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The section Formula for a smooth curve begins as follows:
"Let be an injective and continuously differentiable (i.e., the derivative is a continuous function) function. The length of the curve defined by can be defined as the limit of the sum of linear segment lengths for a regular partition of as the number of segments approaches infinity. This means
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)
The section Formula for a smooth curve contains this statement:
"Let be an injective and continuously differentiable (i.e., the derivative is a continuous function) function. The length of the curve defined by can be defined as the limit of the sum of linear segment lengths for a regular partition of as the number of segments approaches infinity."
But "as the number of segments approaches infinity" is the wrong condition for the sum of the lengths of the approximating segments to approach the length of the curve.
(It is easy to come up with counterexamples where the number of segments approaches infinity, yet the sum of their lengths does not approach the length of the curve: Just keep the length of one segment of length > 0 fixed, while the others become more numerous and approach infinitely many.)
Instead, the correct condition is that the lengths of all the approximating segments of the inscribed polygon must approach zero.
"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)
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)