Talk:Curvature

(Layout problem)

The illustration and the text are interfering with each other, as viewed from Netscape. I've tried putting a colon before the "div", and I've tried putting "br" before and after it, to no avail.
Michael Hardy 20:12 Mar 14, 2003 (UTC)

This seems to be affecting a number of images that used to work correctly in Netscape (they still work as expected in IE). Was something changed in the Wiki software that is affecting this? I'll change it to using a table.
Chas zzz brown 22:50 Mar 14, 2003 (UTC)

Earth's curvature

Seeking information on the Earth's curvature, but no linkage from this page. I've read that "The earth's curvature is not visible from altitudes lower than about 20 miles.", but I'd really like a cite.
~ender 2007-08-21 12:06:PM MST —The preceding unsigned comment was added by 70.167.217.162 (talk)

Signed curvature in three dimensions

It seems noteworthy to me that the local curvature can easily be obtained by adding an obvious term. If one extends the given equation $\kappa ={\frac {|{\dot {r}}\times {\ddot {r}}|}{|{\dot {r}}|^{3}}}$ by the directional vector normalized to unit length the curvature vector becomes as signed quantity: $k_{3D}={\frac {{\dot {r}}\times {\ddot {r}}}{|{\dot {r}}\times {\ddot {r}}|^{3}}}{\frac {|{\dot {r}}\times {\ddot {r}}|}{|{\dot {r}}|^{3}}}$

Where the added term makes $k_{3D}$ consistent to the sign in the signed curvature k for the two dimensional case:

$k={\frac {x'y''-y'x''}{(x'^{2}+y'^{2})^{3/2}}}.$

Thus it is possible to give also a signed curvature for a three dimensional curve. Then one can integrate this and obtain, for example the 'net' curvature for a Lissajous (1:2) figure to be (0.0,0.0,0.0) instead of the unsigned case, where the curvature adds up.

I verified this 'experimentally' in Mathematica. However, can this be found in literature?
User:Aritglanor Friday, June 19, 2009 at 3:44:14 PM (UTC)

About convex and concave curvature

I have made several edits about convex and concave curvature, but those are reverted [1] [2] [3] by User:Sławomir Biały. How sad the Wikipedia only include "convex curvature" for representing "positive curvature" in terms of Gaussian curvature, but not include "concave curvature" for representing "negative curvature"! More sadly, even though the revert might be valid, Wikipedia still not explain "convex curve" nor "concave curve" in terms of Mean curvature... UU (talk) 16:22, 25 December 2016 (UTC)

You're simply wrong that convex/concave surfaces can have negative curvature. A (smooth, strictly) locally convex surface, by definition, is a surface that locally lies on one side of its tangent plane. A neccesary and sufficient condition for local convexity is positive Gauss curvature. Sławomir Biały (talk) 16:58, 25 December 2016 (UTC)

Need revision for "Curvature of Plane curves"

Please see the latest edits by Lichinsol (reverted by some editor) in history. The section "Curvature of plane curves "lacks clear mathematical illustrations and things are written in an unsynchronized manner. Also suggestion for a derivation of" Local expressions " formula on my edit.
Need suggestions!
(The editors who are repeatedly reverting my edits are edit warring. I am not. Without looking into the edit and simply reverting just because it is bold is not the reason to revert as suggested by BRD. BRD too is not the reason to revert. I want to re revert, but then I would be warring.) Lichinsol (talk) 01:36, 2 October 2019 (UTC)

The old version really had problems : using "time" for arc length which is actually "distance".It was also discussed in an old discussion. The present has a derivation for the local expressions too. I would also suggest a derivation for the polar coordinates expression too, if that's possible.VaibhavShinchan (talk) 06:53, 3 October 2019 (UTC)

I cannot understand what this statement means in the Local Expressions :"They can be expressed in a coordinate-independent manner......". Commas are used inside the determinent. This way of writing is probably not used anywhere in the article Determinant. Either a citation should be provided or it should be explained what the determinant actually means.VaibhavShinchan (talk) 07:03, 3 October 2019 (UTC)

@D.Lazard: Please, discuss with each other the disputed edit. — MarkH21 (talk) 07:21, 5 October 2019 (UTC)

The number of modifications made in a single edit by Lichinsol makes difficult to recognize whether this fixes some content issues of the previous version. On the other hand, here are several changes that are not acceptable:

• In section "Curvature of plane curves", replacement of regular prose by a bulleted list. This is against Wikipedia standard, see MOS:LISTBASICS
• Same section: The last bulleted item is nonsensical: it is the curvature that is defined as a rate of change, not the converse as said in Lichinsol's version
• Same section: The last section is indented with "blockquote" without any apparent reason
• Section "Local expression" An useful explanation is removed or hidden in a collapsed box entitled "derivation"
• Section "Curvature of the graph of a function": Lichinsol's heading is badly formatted. Worse, Lichinsol's removes all reference to the graph of a function for introducing a confusion (rather common, I must admit) between a function, which is not a curve, and thus does not has curvature, and its graph, which is a curve and not a function.

I have not checked Linchinsol edits further, but this is sufficient for a revert of the whole edit. If some issues need to be fixed or if some point needs to be improved, please proceed as follows: if the issue or the improvement can be clearly explained in the edit summary, proceed with an explicit edit summary, without any other modification in the same edit. If more detailed explanations are needed, then open a thread in the talk page for given them in details. In any case, respect the Wikipedia rule that asserts that a disputed edit must get a consensus before being kept or redone. D.Lazard (talk) 09:31, 5 October 2019 (UTC)

Here is my say:

First, please open the "archive" of this talk page, and u will find a dispute titled "Meaning of dT/ds", in which someone points out that 'arc length' was written 'time' instead. Writing this way is dubious. This error was not rectified since then. The physical meaning of dT/ds is necessary to be written, but not in this way.

In the section "Curvature of Plane Curves", which explains about dT/ds as the curvature, but it is not written why it is the curvature. In the end paragraph of the section, a small hint of it was given: d(theta)/ds, but it defintely still does not explain the meaning. That is why the section is "unsynchronized" and "lacks information on topic".

Adding bullets to the section should not be a problem at all. It was necessary there.

The "blockquote" was by mistake. I use Visual edit always. Maybe some keyboard key combinations had lead to it being added.

What D.Lazard is talking about of graph of function is incorrect. The subsection was only for, y=f(x). Every function has a graph and may not necessarily be written in the form y=f(x), for example of a circle. So the subsection heading is wrong.

In section "Local Expresssions", the derivation was added by me. It is necessary for explanation, and that hiding it in "show template" makes the article look cleaner, not too harsh to the eyes if the formulae and procedures were thrown naked to the article.

Lastly, I would add that I have not found any weight in almost any of the points mentioned by D.Lazard above. The edit might be large in size, accounting the many problems, they could not be done separately.Lichinsol (talk) 16:24, 5 October 2019 (UTC)

Bullets: It is your right to think that not taking care of Wikipedia rules is not a problem, but if you want to take your part of this encyclopedic project, you must convince other editors that in this case, it makes the article better. Saying "It was necessary there" is not a good way to convince anybody.

The derivation was added by me. It is necessary for explanation: A derivation is a proof, not an explanation. A proof may be useful for supporting an explanation, but can never replace it. Here you have removed an explanation and added a proof. This does not explains anything

Every function has a graph and may not necessarily be written in the form y=f(x), for example of a circle: In this article, all functions are supposed to be differentiable (otherwise, the curvature is not defined); the implicit function theorem says exactly that all differentiable function can be written in the form $y=f(x).$ Also, a circle is not a function, although the upper half circle is the graph of a function. D.Lazard (talk) 17:15, 5 October 2019 (UTC)

The bullets are making the things more clearer. If there are problems, then we may remove the bullets.

For the y=f(x) problem, I think u already know the answer. The sub-section was created for simple functions which can be written in the form y=f(x) and not for others like a circle,Cycloid, etc. The implicit function theorem says that they can be written in the form y=f(x), but the sub-section is only for simple functions where the independent variable(x) can be separated from the dependent variable(y) easily. No too deep thinking in this case. Take the 'simple' word as intuitive as possible here. (Would u prefer to convert the cartesian equation of a cycloid to y=f(x) first or prefer the parametric equation for finding the curvature. Obviously u know the answer. It is almost impossible to convert the cartesian equation of cycloid in the form y=f(x))

The derivation is much better than the explanation written in the present article, and the derivation explains everything, provided the "Curvature of Plane Curves" is read thoroughly. The explanation lacks why an "extra factor of reciprocal of tangent modulus is present in the formula for curvature". Require the attention of D.Lazard in this matter.----Lichinsol (talk) 05:55, 7 October 2019 (UTC)

On the point of bullets and proofs, we should certainly strive for prose over bullet points, as well as prose explanation over derivations and proofs (MOS:PROSE). — MarkH21 (talk) 08:12, 7 October 2019 (UTC)

But the article MOS:PROSE gives the example of a list. What I did was that the bullets were actually prose in themselves. They were not plain 2 to 3 words in a bullet. Every bullet had material in it. The bullets were made for a reason and they clearly do a better work than if prose were used instead.

It is a mathematical article and adding a derivation to it too is for the betterment of the article. Many mathematical articles on wikipedia provide explicit derivations than explanations(See Pendulum (mathematics).----Lichinsol (talk) 13:53, 7 October 2019 (UTC)

I didn't look at the particular content here, I meant as a general principle. But generally, prose is still better than bullets containing prose unless there is a clear reason for the bullets. Derivations certainly have a place on mathematical articles as well, but should not be dominant. We should keep WP:NOTTEXTBOOK in mind. I think D.Lazard has more specific comments on this particular instance. — MarkH21 (talk) 21:16, 7 October 2019 (UTC)

The bullets were made because the section was to elaborate the "number" of possible ways in which curvature could be defined, So making bullets was important. Adding derivations to the article may not make it a textbook necessarily. Many articles on wikipedia provide proofs & derivations. I can't find what problems are being encountered by the editors. Please clearly look into the edit before foretelling that it makes the article a textbook or the bullets are ruling out the norms.Lichinsol (talk) 03:53, 8 October 2019 (UTC)

I have read again the stable version, and also Lichinsol's one. It appears that the stable version is globally correct in the sense that the given formulas are correct and their notation well defined. This is the most important for users who access to this article for remembering technical details that they may have learnt and forgotten. On the other hand the article is highly confusing for a reader who knows nothing about the subject. One of the main issue is that the section heading do not reflect correctly their content. For example "Precise definition" is not about the definition of the curvature, but about a formula for calculating it. "Local expressions" is nonsensical for a concept that is purely local. Here are some examples of confusing assertions in the content: In the first sentence, the use of "loosely" instead of "strongly". In the first section, the wrong implicit assertion that the concept was first introduced by Cauchy (while, he only proved a theorem about the curvature). The fact that the different definitions/characterization concern only differentiable curves is fundamental, but hidden or delayed to a later section (not even linked). I could give many other examples, but this would waste time that would beter used to improve the article.

So I agree with Lichinsol that a complete rewrite is needed. However, his tentative does not address the main issues (except for the emphasis given to Cauchy's characterization). On the other hand, his edit adds some confusion to an article that is already confusing. I have given some example above. Here are two others: the introduction of a {{see also}} template in place of a lacking link in the body of the section. The introduction in the first section of "the domain" without specifying which domain is considered (the same problem occurs in another section of both versions). So Lichinsol's version, does not improve the article, and, is not even a step toward a better article,

I'll try some edits to the article for fixing the main issue. D.Lazard (talk) 13:16, 8 October 2019 (UTC)

Edits done by D.Lazard

Presently, the edits done by D.Lazard has many problems. I am addressing them:

• In section, "Plane Curves" , the first line is incorrect. A point does not have any direction, but a vector has. So the intuitive definition must be rectified.
• The defintion of osculating circle is completely incorrect. A circle can be defined by atleast 3 points. 2 points cannot make a circle.
• Don't add your personal opinions,"the derivative of the unit tangent vector is probably less intuitive than the definition in terms of the osculating circle...". If you can't understand it does not mean that it is less understandable.
• "Convergence" word is used. I have never heard it before. Replace it with another word.

Lichinsol (talk) 10:58, 9 October 2019 (UTC)

Some of your points here seem to be off :

* "A point does not have any direction" : but the derivative of a moving point certainly has, which is clearly what the sentence here refers to.

* "The defintion of osculating circle is completely incorrect. A circle can be defined by atleast 3 points. 2 points cannot make a circle" : if you are refering to the definition in Curvature#Osculating_circle it is completely correct : a circle can be defined by 2 points and a tangent direction at one of them.

The two other, which do not touch on technical matters, seem pertinent. If you want to improve the article you should probably stick to comments like these. jraimbau (talk) 11:27, 9 October 2019 (UTC)

The derivative of a moving point does not have any direction. It is senseless. What here is to be written is that "the rate of change in the direction of tangent vector is defined as curvature." I don't seem to visualize that the derivative of a point can have any direction, nor I have seen these anywhere in literature. If possible, provide some evidence. Lichinsol (talk) 11:51, 9 October 2019 (UTC)

The derivative of a moving point is a vector, and as such has a direction. More precisely, the derivative of a moving point is defined as a limit involving a difference of two points, and in a Euclidean space, as well as in any other affine space, the difference of two points is defined and is a vector (while the sum of two points is not defined); see Affine space. D.Lazard (talk) 12:43, 9 October 2019 (UTC)

I don't find meaning in what D. Lazard is saying. Take a space and 2 distinct points in it. The difference b/w the 2points is obviously a vector. But it does not tell about the direction b/w the 2 points. Give me a perfect evidence for it . I still don't believe it.Lichinsol (talk) 13:19, 9 October 2019 (UTC)