Talk:Gaussian curvature

What's the relationship between Gaussian curvature and Ricci scalar? --Itinerant1 20:19, 30 March 2007 (UTC)
 * Have a look at Curvature of Riemannian manifolds, which explains a bit more. --Salix alba (talk) 22:18, 30 March 2007 (UTC)

Mistake on the First Line
Gaussian and Gauss curvature are not the same thing. There is an idea of Gaussian curvature: which this article is about. There is also an idea of Gauss-Kroneker curvature TENSORS (see http://www.springerlink.com/content/j407l35440653537/ ). The two things are not the same! —Preceding unsigned comment added by Dharma6662000 (talk • contribs)

Figure containing hyperboloid, cylinder, and sphere
It may be very obvious, but does anybody feel it should be made clear in the figure caption that the colormap does not represent Gaussian curvature? — Preceding unsigned comment added by Dfsp spirit (talk • contribs) 08:13, 25 April 2019 (UTC)

Paper
Am I the only one whom the "Paper" section strikes as super-pedantic and silly?67.172.93.9 06:15, 4 December 2007 (UTC)

error in alternative definition?
The formula in terms of the connection appearing at the beginning of the "alternative definitions" section may not be correct, at least if one defines e_1 and e_2 the usual way. If their commutator is nonzero, aren't we missing a term here? Tkuvho (talk) 09:27, 12 April 2010 (UTC)

Better Classification
There should be three sections in the article, each talking about negative, positive or zero Gaussian Curvature, with different examples and what that means with respect to the principal curvatures and vectors. Basically, there should be more insightful discussion with Gaussian Curvature as it has very far reaching consequences in differential geometry. —Preceding unsigned comment added by 128.100.86.103 (talk) 23:09, 19 April 2011 (UTC)

This article needs to explain Gaussian curvature to non-mathematicians as well
I was just telling a non-mathematical friend about Gaussian curvature, and thought that maybe referring her to this Wikipedia article might help her understand it.

But with the exception of the three pictures of surfaces of negative, zero, and positive curvature, the article is total gobbledygook to a non-mathematician.

If the article were supposed to be accessible solely to mathematicians, it would be OK, though it could still be improved with more illustrations (such as of the principal curvatures of various surfaces in 3-space, including some intrinsic surface shown in various isometric embeddings in 3-space).

But Gaussian curvature and the Theorem Egregium are such basic and beautifully simple concepts, that there is no reason on earth that an introductory section could not explain these both so that non-mathematicians could easily understand them.

Even the amazing Gauss-Bonnet theorem could be explained in layman's terms in a way that non-mathematicians could get the rough idea of it.

That this article is almost 100% inaccessible to laymen is a crying shame.Daqu (talk) 21:19, 1 April 2012 (UTC)
 * Such an explanation can be found in the Gaussian_curvature section. Tkuvho (talk) 15:00, 5 April 2012 (UTC)
 * That section
 * We represent the surface by the implicit function theorem as the graph of a function, f, of two variables, in such a way that the point p is a critical point, i.e., the gradient of f vanishes (this can always be attained by a suitable rigid motion). Then the Gaussian curvature of the surface at p is the determinant of the Hessian matrix of f (being the product of the eigenvalues of the Hessian). (Recall that the Hessian is the 2-by-2 matrix of second derivatives.) This definition allows one immediately to grasp the distinction between cup/cap versus saddle point.
 * can't be said to be aimed at non mathematicians. It got a lot of jargon: critical point, implicit function theorem, gradient, Hessian, eigenvalues, its probably only understandable by someone well versed in differential geometry. There are other ways of describing it Koenderink tries hard, I think based on how the normal vector change.--Salix (talk): 16:50, 5 April 2012 (UTC)


 * Anyone can easily see that, as Salix points out, that section is indeed full of mathematical jargon. The best way to explain Gaussian curvature (at a point P) to the layman is to describe the principal curvatures -- the maximum and minimum curvatures of the intersection of a plane through the normal vector at P, over all the 180° of rotational positions such a plane can take -- and define Gaussian curvature (of the surface at P) as the product of these curvatures. Several good illustrations would make this clear, as long as a) the idea of the curvature of a curve is clear, and b) as long as it is clear that these two curvatures should be taken with opposite sign if they bend in opposite directions.


 * This refers to a surface embedded in 3-space. (The layman doesn't need to know about the intrinsic Gaussian curvature of an abstract Riemannian surface.)Daqu (talk) 04:01, 15 May 2012 (UTC)


 * Note that the term "sectional curvature" is not usually used to refer to the curvature of the curve obtained as the intersection of the surface with a plane containing the normal direction. Tkuvho (talk) 16:38, 15 May 2012 (UTC)


 * Good point -- that usage should be changed in what looks like an otherwise good Informal definition section..Daqu (talk) 07:38, 18 May 2012 (UTC)

Implicit form
The article has for the $$F(x,y,z)=0$$


 * $$K=\frac{[F_z(F_{xx}F_z-2F_xF_{xz})+F_x^2F_{zz}][F_z(F_{yy}F_z-2F_yF_{yz})+F_y^2F_{zz}]-[F_z(-F_xF_{yz}+F_{xy}F_z-F_{xz}F_y)+F_xF_yF_{zz}]^2}{F_z^2(F_x^2+F_y^2+F_z^2)^2}$$

With the comment "This one is too long, cannot figure out how to divide it nicely". The reference comes from Mathworld http://mathworld.wolfram.com/GaussianCurvature.html, which references a work by Trott. This is a little unsatisfactory as there is a $$F_z^2$$ factor on the bottom and ideally we should see symmetry in the three coordinates.

I've found a nicer reference which presents a different form. $$ K=\frac{ \nabla F\ H^{*}(F)\ \nabla F^{\mathsf T} }{ |\nabla F|^4 } $$ Where $$\nabla F$$ is the gradient and $$H^{*}(F)$$ is the adjoint of the Hessian. It also references Spivak who has perhaphs the nicest presentation

$$ K=-\frac{ \begin{vmatrix} H(F) & \nabla F^{\mathsf T} \\ \nabla F & 0 \end{vmatrix} }{ |\nabla F|^4 } $$

or explicitly

$$ K=-\frac{ \begin{vmatrix} F_{xx} & F_{xy} & F_{xz} & F_x \\ F_{xy} & F_{yy} & F_{yz} & F_x \\ F_{xz} & F_{yz} & F_{zz} & F_x \\ F_{x} & F_{y} & F_{z} & 0 \\ \end{vmatrix} }{ |\nabla F|^4 } $$ I've checked the formula and they expand to the same thing, the $$F_z^2$$ does cancel nicely.--Salix alba (talk): 20:33, 14 April 2015 (UTC)


 * Just wondering: Is that right-hand column of the matrix of the numerator determinant correct, with three Fx's? 2601:200:C082:2EA0:D8BD:7D54:9940:6DDD (talk) 23:39, 21 April 2023 (UTC)

Obscure passage
The second point in the 'informal definition' is very obscure. It uses the phrase 'plane orthogonal to the surface' without explaining what this means. If it is the same as a 'normal plane' (which has already been defined) it is unnecessary and potentially confusing, while if it is different from a normal plane, it is even more important to say what it is. The passage then talks about rotating the orthogonal plane in two directions, without saying what they are - clockwise and anticlockwise? - and asserts that 'the normal curvatures will be zero'. I can see that if a value goes continuously from positive to negative, it must be zero somewhere, but the plural 'curvatures' implies more than one position of zero curvature, which is either wrong or not clearly explained.2A00:23C5:6487:4701:845C:797:3656:22D7 (talk) 11:59, 24 June 2020 (UTC)

Bad writing
This sentence, in the section Informal definition, is an example of bad writng:

"Because one principal curvature is negative, one is positive, and the normal curvature varies continuously if you rotate a plane orthogonal to the surface around the normal to the surface in two directions, the normal curvatures will be zero giving the asymptotic curves for that point."

"Will be zero"??? When? Where? What do you mean?

What you mean is that for at least two directions (and four for non-umbilic points), the normal curvature will equal zero. And that those are the asymptotic directions.

But if you don't have the time or inclination to write clearly, you're just making more work for someone else.

Problematic caption
The caption accompanying one illustration, depicting two surfaces, reads as follows:

"Two surfaces which both have constant positive Gaussian curvature but with either an open boundary or singular points."

One surface resembles the outer half of a torus of revolution, the shape of a wrist bangle, while the other resembles an (American) football. The two non-smooth points of the football must be the "singular points" referred to. But it is not clear what "open boundary" has to do with the bangle.

This section needs to give the metric to say what E, F, and G are, which are extensively used but never defined, and also say what L, M, and N are in the first formula. — Preceding unsigned comment added by Donpage (talk • contribs) 15:51, 12 April 2024 (UTC)