Talk:Curvature/Archive 1

Archive 1

(Circle's curvature)

Is the curvature for a circle really 1/r? I would guess this is something someone writing an article on curvature wouldn't get wrong, but it doesn't work out for me...

$f=x^{2}+y^{2}-r^{2}$

$\nabla f={\begin{pmatrix}{2x},{2y}\end{pmatrix}}^{T}$

$\kappa _{f}=\nabla \cdot \left({\frac {\nabla f}{\|\nabla f\|}}\right)=$

$\nabla \cdot {\frac {{\begin{pmatrix}{2x},{2y}\end{pmatrix}}^{T}}{\sqrt {4x^{2}+4y^{2}}}}$ $=$ $\not =$ here it is

${\frac {(2+2)}{2{\sqrt {r^{2}}}}}=$

${\frac {2}{r}}$ !

— Preceding unsigned comment added by 128.174.244.253 (talk) 04:15, 17 January 2006 & Tosha (talk • contribs) 23:58, 20 January 2006 Tosha (talk • contribs) 23:58, 20 January 2006

Yep, the curvature of a circle is 1/r! The mistake in your algebra is where you calculate the divergence - you have assumed the denominator is constant, when it isn't.
Kaplin 21:14, 19 July 2006 (UTC)

Curvature in polar coordinates

The given expression for curvature when using polar coordinates is confusing. If we are using polar coordinates, then generally the curve will have the form $r=r(\theta )$ so one should expect an expression for the curvature to be in terms of the derivative of r with respect to $\theta$. Writing it in terms of F(y) is confusing.
--81.153.87.195 15:37, 30 January 2007 (UTC)

Corrected this.
--Bob The Tough 13:10, 12 February 2007 (UTC)

Numerical evaluation of curvature

I was looking for a method to compute surface curvature at a point on a surface in 3D, given a number of points in the neighborhood. I couldn't find any. Should this be added to this article, or should it link to some more general article on numerical methods for e.g. numerical differentiation / integration ?
Mauritsmaartendejong 20:26, 19 June 2007 (UTC)

Maybe it would be enough to add something like "Here y' is derivative of function y." after the first equation? Then one would find link to "Numerical differentiation" in the article "Derivative".
--Martynas Patasius 12:56, 20 June 2007 (UTC)

This might be the case where you have to go to the research litrature. I've seen several techniques used: for example fitting a patch and then calculating it from the equations from the patch, there are several other techniques[1]. I'm not aware of any one method which is superiour to others and they can vary depending on the type of data you have.

For this reason I don't think it wise to explicitly mention any one method.
--Salix alba (talk) 17:43, 20 June 2007 (UTC)

Curvaton in Physics

Please, may someone explain what is a Curvaton in physics. I've been trying to reach information about the Curvaton in Wikipedia, but I was unable to find it.
— Preceding unsigned comment added by 84.122.179.77 (talk) 21:36, 2 January 2008 (2 edits) According the dictionary, a Curvaton is:

1. (cosmology) an scalar field that can generate fluctuations during inflation, but does not itself drive inflation; it generates curvature perturbations at late times after the inflaton field has decayed and the decay products have redshifted away, when the curvaton is the dominant component of the energy density

In someone in physics could create a full article, I think is very interesting.
— Preceding unsigned comment added by 84.122.179.77 (talk) 21:38, 2 January 2008

You could try Wikipedia:Reference desk/Science, which is the best place for science questions.
--Salix alba (talk) 22:19, 2 January 2008 (UTC)

arclength parametrisation

When parametrising using arclength, the formulas for curvature become nice. Maybe this should be added.
Randomblue (talk) 13:41, 4 February 2008 (UTC)

A more immediate issue is the redlink arclength parametrisation (or arclength parametrization). Anyway, I think much of the article should be moved out to curvature of a curve, and much more detail should be supplied there. A survey of various kinds of curvature should remain here. I tried to do this at one point, but became very badly stuck trying to discuss curvature in general in a way that would cover all cases.
Silly rabbit (talk) 13:47, 4 February 2008 (UTC)

Osculating circle picture

It's nicely done (in colors!), but is quite misleading: it shows the osculating circle at a vertex of the curve, where one of them is contained inside the other. At a generic point of non-zero curvature, the osculating circle will actually cross the curve in a bent version of the cubic parabola meeting the x-axis at the origin. Is there anyone knowledgeable about graphics generation who can fix the picture?
Arcfrk (talk) 23:33, 22 February 2008 (UTC)

Article needs to split

One of the big differences between a dictionary and an encyclopedia is that each topic has its own page. The three different definitions of curvature in the article are from pretty different domains.

I'm of the opinion that this article should become 3 articles, maybe Curvature (space) Curvature (surfaces) and Curvature (curves).
- (User) WolfKeeper (Talk) 05:06, 27 April 2008 (UTC)

No, the concept is the same, what if one wants to understand "what curvature is".
--Tosha (talk) 05:01, 29 April 2008 (UTC)

Bravo, spoken like a true realist. An idealist would ask how many liftings of any given intrinsic definition are there to an extrinsic definition in an embedding space of a given higher dimension. If exactly one then "the concept is the same" becomes reasonable, but this is only true in certain cases. While I'm not aware of any general characterizations of the exactly-one case, this means nothing as differential geometry is not my area (I'm more of a computer scientist with algebraic leanings). That said, I'd still love to know more about the state of the art concerning that question.
--Vaughan Pratt (talk) 03:07, 7 December 2008 (UTC)

Curvature of plane curves

The formula for the curvature of a plane curve is assuming that T' is at 90degrees to T.

For example, if gamma(s)=(0.0,s**2) then T=(0,2) which gives a radius R=0.5, but this is a straight line, x=0.0
—Preceding unsigned comment added by Steve 33025 (talk • contribs) 6 May 2009

As T is of unit length then T' must be at 90 degrees to T. This can easily be proved as for T to be unit length T.T=1 differentiating gives 2 T'.T=0 so T' has no component in T direction. It might be worth mentioning this in the article.
--Salix (talk): 17:12, 6 May 2009 (UTC)

Then i guess my issue is with the equation

$T(s)=\gamma '(s)$

I can't see why the first derivative $\gamma '(s)$ should be of unit length
—Preceding unsigned comment added by Steve 33025 (talk • contribs) 7 May 2009

If you look closely at the article the curve gamma is chosen to be parameterised by arc length which implies that it of unit speed, so $|\gamma '(s)|=1$
--Salix (talk): 09:16, 7 May 2009 (UTC)

OK, makes sense to me now, thanks
— Preceding unsigned comment added by Steve 33025 (talk • contribs) 09:14, 12 May 2009‎

Intrinsic and extrinsic curvature

As it stands, the article is almost entirely about extrinsic curvature, derived from paths in a two or three dimensional space. I think this article needs more material relating to intrinsic curvature which is of great importance in differential geometry and its applications in physics such as general relativity. The distinction is a major one. An extrinsic curvature may be calculated for the orbit of a planet at a particular point in its path, but the intrinsic curvature of space-time due to the gravitational field causes the orbit to have the form it does (including things like the precession of the orbit of mercury).
Elroch 00:48, 13 February 2006 (UTC)

Agree that we could do with a treatment of intrinsic curvature and that the distinction is a major one. Maybe, a distinction made upfront with some emphasis throughout.
--Eddie | Talk 10:27, 13 February 2006 (UTC)

It is elementary article, there is something about Gauss curvature, and if someone needs more there are refs.
Tosha 00:13, 16 February 2006 (UTC)

I agree. I added a more general introduction after I made my comment. I hope this helps.
Elroch 22:57, 16 February 2006 (UTC)

I have a problem with the paragraph "For example, an ant living on a sphere could measure the sum of the interior angles of a triangle and determine that it was greater than 180 degrees, implying that the space it inhabited had positive curvature. On the other hand, an ant living on a cylinder would not detect any such departure from Euclidean geometry; the cylinder has extrinsic curvature, but no intrinsic curvature." Gauss curvature is intrinsic, so it can be determined by either ant, no matter what kind of sphere it lives on. The only difference between the sphere and the cylinder is that the sphere has Gauss curvature different from zero and the cylinder does not. The current formulation makes the impression that the ant on cylinder cannot determine the curvature, which is wrong.
Anša (talk) 18:09, 5 September 2009 (UTC)

I agree, the ant explanation in the article is wrong. Given that Gaussian curvature is intrinsic, it is measurable on any 2D surface, and as Anša says, that includes cylinders, where it happens to be zero. The trick is more like this: Flat ants living on a 2D surface unaware of bending in the third dimension nonetheless try to measure it using points on their surface, rulers and protractor. Using only two points there's no way for them to detect the difference between positive, negative curvature or flat (remember, their rulers measure only along their surface), hence they don't find little k, kmax, kmin and similar.

However, when the ants measure triangles, they do indeed see the difference between various degrees of curvature (angles may not add up to 180 degrees), hence they can detect the overall effect of curvature in their X and Y directions, even if they can't tell which direction is the curviest. This is what Gaussian curvature gets at, and why it's said to be intrinsic.
Gwideman (talk) 01:24, 11 March 2010 (UTC)

Eigenvalues of second fundamental form

AFAIK k1 and k2 are not eigenvalues of second fundamental form. At the very least this needs an explanation.
Here's a paper on how to get k1 and k2: http://www.cs.berkeley.edu/~sequin/CS284/TEXT/diffgeom.pdf. See equation 56, which uses terms of the first fundamental form (E,F,G) as well as the second (L,M,N). —Preceding unsigned comment added by 66.195.165.72 (talk) 22:11, 17 December 2010 (UTC)

I have included a better description of how to get the principle curvatures from the second fundamental form as well as the shape operator.
Sławomir Biały (talk) 16:25, 26 December 2010 (UTC)

Moved from article

I moved the following from the article as out of place. If it should be in the article at all, it needs a different home, perhaps in a (not yet written) section on applications.
Sławomir Biały (talk) 15:50, 31 December 2010 (UTC)

The magnitude of curvature at points on physical curves can be measured in diopters (also spelled dioptre) — this is the convention in optics. A diopter has the dimension ${{\mathit {Length}}^{-1}}.$ — Preceding unsigned comment added by Sławomir Biały (talk • contribs) 15:50, 31 December 2010‎(2 edits)

add formula for curvature of a surface in form z=f(x,y)

a general surface which occurs many times in physics has the form z=f(x,y) you can add the formula for curvatures kx & ky to the page to make it more complete— Preceding unsigned comment added by 207.6.122.173 (talk) 18:07, 15 January 2012‎

We aren't tabulating formulas for the Gauss curvature in this article. Those appear in the main Gauss curvature article.
Sławomir Biały (talk) 18:33, 15 January 2012 (UTC)

Intuitive ants?

"intuitively, this means that ants living on the surface could determine the Gaussian curvature"
I'm not sure that such capabilities of ants are very intuitive!
— Preceding unsigned comment added by 82.10.109.67 (talk) 20:11, 12 July 2012 (UTC)

Cleanup required in section "Curvature from arc and chord length"

The section on "Curvature from arc and chord length" is rather spectacularly bad. Cleanup and references would be most helpful.
Sławomir Biały (talk) 14:28, 27 December 2010 (UTC)

I have substantially reduced the size of the section. Most of this was written in an inappropriately bombastic tone, with very little useful content. A good reference is definitely needed to give context to the stated result.
Sławomir Biały (talk) 16:20, 31 December 2010 (UTC)

For the sake of lowering barriers to verifying good faith, i note that a colleague made a signed contribution at this location on the page, at 11:13, 8 January 2013, and removed it at 14:30 the same day. Having read the content in context, i'll be surprised if anyone should consider further attention appropriate.
--Jerzy•t 09:24, 13 July 2013 (UTC)

Meaning of dT/ds

This passage seems to take a wrong turn in explaining the relationship dT/ds:

Another way to understand the curvature is physical. Suppose that a particle moves along the curve with unit speed. Taking the time s as the parameter for C, this provides a natural parametrization for the curve. The unit tangent vector T (which is also the velocity vector, since the particle is moving with unit speed) also depends on time. The curvature is then the magnitude of the rate of change of T. Symbolically,

$\kappa =\left\|{\frac {d\mathbf {T} }{ds}}\right\|.$

I studied physics, not math, and there I was taught that the letter "s" represented distance, not time; the "s" coming from the Latin spacium. Taking "s" as distance suggests a simpler, yet non-dynamical, explanation of the equation.

Since no citations are given for the explanation, could someone provide some source for this usage. SteveMcCluskey (talk) 13:26, 27 April 2014 (UTC)

Yes, s represents the arclength parameter along the curve. How is that inconsistent with what is written? Sławomir Biały (talk) 14:17, 27 April 2014 (UTC)

Signed Curvature in 2 dimensions

I came here just to check this simple point, but found the explanation unhelpful, and the animation worse than useless- it just made me dizzy. Its clever, but too clever - just baffling. I gave up and asked a friend. He told me that the sign of the curvature is the same as that of the 2nd derivative at the point in question. Now !that! was simple, straightforward, and helpful. — Preceding unsigned comment added by 77.96.60.31 (talk • contribs) 2014-12-05T17:36:00

... and, unfortunately, incorrect. The sign of the curvature of a curve in a plane depends on the orientation of the curve: a choice of direction along the curve. The second derivative is also not necessarily defined: for that, you need to nominate an x-axis, which for a general curve in a plane, is not a natural (i.e. unambiguous) choice. —Quondum 00:12, 6 December 2014 (UTC)

I think for a graph, it's safe to assume that one specifically has the induced orientation in mind. Sławomir Biały (talk) 02:49, 6 December 2014 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Curvature/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

needs refs, try finding some here. Cronholm144 15:03, 18 May 2007 (UTC) Also deals primarily with the curvature of curves and Gauss curvature. Should offer a more generic elementary interpretation of curvature, pointing to other articles for details. Silly rabbit 19:13, 21 May 2007 (UTC)

Last edited at 22:11, 28 May 2007 (UTC). Substituted at 01:56, 5 May 2016 (UTC)

About convex and concave curvature

I have made several edits about convex and concave curvature, but those are reverted [2] [3] [4] 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)

The concept being described is basically the tangent vector, which is the first derivative of a parametric smooth curve (i.e. a "moving point"). You can find plenty of sources at the article there or take any multivariable calculus textbook. — MarkH21 (talk) 05:30, 10 October 2019 (UTC)

The problem is not as much about circles drawn through P and Q as about their limiting “circle” which can degenerate to a straight line. That article does not address the problem either, which is a serious shortcoming. Incnis Mrsi (talk) 18:06, 15 October 2019 (UTC)

Section "Examples: Examples: Rate of Change of the Unit Tangent Vector

Apparently, this section is aimed to explain curvature on few examples. Unfortunately, it uses a terminology that differs from the remainder of the article ("Rate of Change of the Unit Tangent Vector" vs "curvature", without explaining the relationship between the two terminology; it uses, without any definition, measurement units that are not considered elsewhere in the article; most sentences are incomplete; etc.

Therefore, this section is of no help for understanding the remainder of the article. A list of simple examples could be useful, but writing it from scratch is definitively easier than starting from this section. I'll thus remove this section per WP:TNT. D.Lazard (talk) 09:40, 4 December 2020 (UTC)

@D.Lazard: The section looked was a jumbled mess (link to old version), so that sounds more than reasonable. — MarkH₂₁^talk 12:17, 4 December 2020 (UTC)

curvature reciprocal radius

Correction: The curvature of the circle is not the " exact " reciprocal of its radius, it is not mathematically expressed without an irrational number that is not an exact integer.

Mathematics is stable and called an exact science only according to a single expression that defines exactitude (exactness), namely:

a / a = 1 197.149.243.216 (talk) 13:31, 12 January 2022 (UTC)

zionion.blogspot.com 197.149.243.216 (talk) 13:32, 12 January 2022 (UTC)

Curvature of a surface

It is well known in schoolkid geometry that a sheet of paper can adopt single-order curves (zero Gaussian curvature) such as conical and cylindrical forms, but not second-order curves (non-zero Gaussian curvature) such as spherical or hyperbolic. This appears to be a rather drastic omission from an articled titled simply "Curvature". Either this needs adding or, if it is treated in another article, than it needs a prominent link that even readers like me can find. — Cheers, Steelpillow (Talk) 13:47, 9 February 2022 (UTC)

The fact that a smooth deformation of a paper sheet has a zero Gaussian curvature everywhere is a corollary of Gauss' Theorema Egregium, which is mentioned in Gaussian curvature. There is no reason to mention this here. I do not know what do you mean by "adopting curves of some type" and what you call the order of a curve. Nevertheless this is far from "schoolkid geometry", as Theorema Egregium is a theorem of Riemannian geometry that can not be taught at elementary level. Moreover, it is known that every smooth deformation of a paper sheet is a ruled surface, but this is a harder theorem. D.Lazard (talk) 14:52, 9 February 2022 (UTC)

Well, I learned it at school around the age of 12, when our teacher introduced us to hyperbolic paraboloids as ruled surfaces. I was reminded of it again at 15 when it was the turn of spherical trigonometry. By 16 my Geometrical Drawing exam syllabus was requiring me to draw up surface developments approximating quadrics. Just because the underlying theorems are not taught, does not make the whole topic abstruse. I was at a perfectly ordinary school taking perfectly ordinary exams, nothing special there. So let me pose the issue a different way. A kid comes across the idea that you can roll paper into a cylinder but not into a sphere. They want to know more, so they turn to an encyclopedia and look up curvature; maybe it's this encyclopedia and they have the brains to look up "surface curvature"; that redirects them to Curvature#Surfaces, which makes no mention of any such distinction. How are you going to help that kid find what they want? — Cheers, Steelpillow (Talk) 16:12, 9 February 2022 (UTC)

To clarify: by "adopt" I mean "be deformed into, through simple physical bending". The distinction I am making is between this - what Mathworld calls a developable surface - and a surface which is not developable. — Cheers, Steelpillow (Talk) 19:11, 9 February 2022 (UTC)