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Mathematics rating:
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 Field: Geometry
One of the 500 most frequently viewed mathematics articles.
This article has comments.

(Layout problem)[edit]

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)

(Circle's curvature)[edit]

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 (talk) 04:15, 17 January 2006 & Tosha (talkcontribs) 23:58, 20 January 2006 Tosha (talkcontribs) 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)

Intrinsic and extrinsic curvature[edit]

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)

Curvature in polar coordinates[edit]

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.
-- 15:37, 30 January 2007 (UTC)

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

Numerical evaluation of curvature[edit]

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)

Earth's curvature[edit]

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 (talk)

Curvaton in Physics[edit]

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 (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 (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[edit]

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[edit]

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[edit]

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[edit]

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 (talkcontribs) 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


I can't see why the first derivative \gamma'(s) should be of unit length
—Preceding unsigned comment added by Steve 33025 (talkcontribs) 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 (talkcontribs) 09:14, 12 May 2009‎

Signed curvature in three dimensions[edit]

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)

Eigenvalues of second fundamental form[edit]

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: 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 (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)

Cleanup required in section "Curvature from arc and chord length"[edit]

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.
--Jerzyt 09:24, 13 July 2013 (UTC)

Moved from article[edit]

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 (talkcontribs) 15:50, 31 December 2010‎(2 edits)

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

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 (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?[edit]

"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 (talk) 20:11, 12 July 2012 (UTC)

Meaning of dT/ds[edit]

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)