Talk:Curvature

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the next derivative

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]

Problematic description

The section Gaussian curvature begins as follows:

"In contrast to curves, which do not have intrinsic curvature, but do have extrinsic curvature (they only have a curvature given an embedding), surfaces can have intrinsic curvature, independent of an embedding. The Gaussian curvature, named after Carl Friedrich Gauss, is equal to the product of the principal curvatures, k1k2."

So: Immediately after the reader is told that Gaussian curvature is intrinsic to a surface, it is defined extrinsically.

This strikes me as rather confusing to the reader.

I hope someone knowledgeable about this subject can fix this. — Preceding unsigned comment added by 2601:204:F181:9410:C541:848E:89D8:1A39 (talk) 04:02, 19 May 2024 (UTC)[reply]