Talk:Principal curvature

Definition needs expert help

There seems to some issues with the definition given. First, I would assume that the principal curvatures of a plane are both 0. The definition would then imply that all curves on the plane would have a curvature between 0 and 0, in other words all curves on the plane would have curvature 0 and would therefore be lines. If one applies the definition given the one of the principal curvatures will always be infinity since one can draw arbitrarily small circles (or approximations thereof) through a given point, this is clearly not what is meant.

Second, the use of maximum and minimum in the definition cannot be extended to higher dimensions. A hypersurface in 4-space would have three principal curvatures but a maximum and minimum definition can only give two of them.

Third, it seems to me that care needs to be taken to get the correct signs for the principal curvatures relative to the orientation of the surface. There is a significant difference in the local shape of the surface between when the principal curvatures have the same or opposite signs.

I've checked MathWorld for a better definition but theirs is rather abstruse. RDBury (talk) 18:48, 16 February 2008 (UTC)

• Strike the third point, on rereading the next paragraph it seems clear now. First and second points still need attention. RDBury (talk) 19:03, 16 February 2008 (UTC)

Yes I think the definition is a little inprecise. Principal curvatures are the min and max of the sectional curvatures. That is if you cut the surface by a plane containing the normal vector and then take the curvature of the resulting curve that will give a sectional curvature. Equiviently we can a curves whose tangent is in the direction of the plane and whose normal vector (in the sense of Frenet-Serret formulas) is the surface normal. Circles in the plane do not satisfy this condition.

See Curvature of Riemannian manifolds for higher dimensional cases.--Salix alba (talk) 00:20, 17 February 2008 (UTC)

Not just imprecise, but wrong. The formal details are in the Darboux frame article, though I am not sure of an easy way to incorporate any of this here. Silly rabbit (talk) 03:39, 17 February 2008 (UTC)

I've rewritten the definition, I hope it is more intelligible (and precise) than the previous version. And I hope in so doing I have answered the points raised above. Simplifix (talk) 21:21, 17 February 2008 (UTC)

Well, I had already made it a bit more precise. But thanks for separating out the "formal definition" from the lead paragraph. I'm not sure I would call this a very good definition, but there is room for expansion. Silly rabbit (talk) 21:26, 17 February 2008 (UTC)

What you have done now is definitely an improvement Silly rabbit. And you have included the piece about the second fundamental form which makes it all clearer. Thanks Simplifix (talk) 10:40, 18 February 2008 (UTC)

The problem with the definition was that it was totally wrong. See this edit where I corrected the definition prior to your arrival here. I agree that your intervening edits were an improvement, but they did not directly address RDBury's concern — the definition was wrong and RDBury spotted it (thanks). So I would say that my original edit was also definitely an improvement as well: replacing wrong maths with right maths generally is. (You seem to be implying that it was not an improvement.) Silly rabbit (talk) 13:34, 18 February 2008 (UTC)

Thanks for the quick response. The new definition seems to work. I was hoping that the formal definition would be a bit more self contained since not everyone who might land on this page will be familiar the fundamental forms (I wasn't at least). It appears however that the concepts are intrinsically linked so it makes more sense the way it is. 68.40.42.254 (talk) 05:07, 25 February 2008 (UTC)

Double bars

Why is the Pi encosed in double bars in "formal Definition", it looks like the norm, but it is just a matrix. Is it just renerered wrong on my browser?Billlion (talk) 07:47, 18 March 2008 (UTC)

Yes, it is a matrix. This is a bad notation. Feel free to change it to something else. silly rabbit (talk) 12:03, 18 March 2008 (UTC)

What is the principal curvature?

Silly Rabbit, would you please here, in the talk, explain to me what are k's appearing in the article? How do you define them? What is the curvature of a sphere with radius R? For this make sure you use what is presented in the article. If you have difficulty, then you may refer to this link. Thanks. Zitterbewegung ^Talk 03:36, 13 October 2008 (UTC)

I think if you would have taken the time to read the article before "improving" it, you would already have your answer:

At each point p of a differentiable surface in 3-dimensional Euclidean space one may choose a unit normal vector. A normal plane at p is one that contains the normal, and will therefore also contain a unique direction tangent to the surface and cut the surface in a plane curve. This curve will in general have different curvatures for different normal planes at p. The principal curvatures at p are the maximum and minimum values of this curvature.

Which is essentially what your version said, but with a better style of overall presentation, I think. Furthermore, this appeared directly below your proposed addition. The paragraph after this one then goes on to describe briefly how to compute the curvature of a curve. Of course, this article isn't about computing the curvature of a curve, so it uses a certain amount of summary style in doing so. Since you seem to be concerned that the article doesn't directly define k_i, I have added this to the existing paragraph. siℓℓy rabbit (talk) 03:54, 13 October 2008 (UTC)

To answer the second part of your question, let M be a sphere of radius r. A normal plane contains the center of the sphere, and so cuts out a great circle (of radius r). The curvature of that circle is 1/r. This is independent of the normal plane chosen, and so the maximum and minimum values of the curvature are both equal to 1/r: that is k₁=k₂=1/r. Any other questions? siℓℓy rabbit (talk) 03:57, 13 October 2008 (UTC)

Now for a challenge of my own: explain in detail how what you wrote is different from the existing definition of the principal curvatures given in the article. If it is the same, then why is there a need to have two almost identical definitions back to back in the article? Thanks for addressing it, siℓℓy rabbit (talk) 04:02, 13 October 2008 (UTC)

Joachimsthal theorem

I have reverted a paragraph recently added by an IP user about Joachimsthal theorem, for the following reasons. Firstly, it is not clear if this very technical theorem is notable enough for appearing in WP. To validate its notability, at least one reliable secondary source must be provided. Secondly, if this theorem deserve to be cited in this article, this is certainly not in the middle of this section, but rather in a section, near the end of the article, entitled, for example, "Further properties".

Apparently, Joachimsthal theorem has been inserted only for introducing a recent result (2014), which is not yet published in a referred journal. A such this is original research and is not allowed in WP.

D.Lazard (talk) 18:22, 30 April 2014 (UTC)