Talk:Curvature

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(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)

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 70.167.217.162 (talk)

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)

Curvature of a surface[edit]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

the next derivative[edit]

If (signed) curvature of a plane curve is the first derivative of tangent angle with respect to arc length, is there a common word for the next derivative, i.e. the first derivative of curvature? —Tamfang (talk) 19:19, 24 January 2023 (UTC)[reply]

Torsion of a curve? –jacobolus (t) 20:59, 24 January 2023 (UTC)[reply]

I now bolded the thing you may have missed. —Tamfang (talk) 21:10, 24 January 2023 (UTC)[reply]

The next-derivative analog of curvature (a naturally bivector-valued quantity) is torsion (a naturally trivector-valued quantity). In the plane of course torsion vanishes (any wedge product of 3 coplanar vectors is 0). You can come up with various other planar concepts involving higher derivatives, but IMO they aren’t really natural analogs of curvature. –jacobolus (t) 03:00, 25 January 2023 (UTC)[reply]

Raph Levien's thesis has a lot of analysis about changes in curvature with respect to arclength, but I am not sure if there are any specific names like what you are looking for. –jacobolus (t) 03:08, 25 January 2023 (UTC)[reply]

Heh, reading Levien's work prompted the question. —Tamfang (talk) 00:19, 30 January 2023 (UTC)[reply]

Next time I see him, I’ll try to remember to ask if there’s a name for this. No promises though. –jacobolus (t) 02:34, 30 January 2023 (UTC)[reply]