Talk:Principal curvature
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[edit] 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)
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- 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)
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- 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)
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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)
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- 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)
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- 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)
[edit] 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)
