Talk:Frenet–Serret formulas

'Formulas in n dimensions' section is misleading

The equation:

${\overline {\mathbf {e} _{2}}}(s)=\mathbf {r} ''(s)-\langle \mathbf {r} ''(s),\mathbf {e} _{1}(s)\rangle \,\mathbf {e} _{1}(s)$

is valid for any arbitrary parameterization. For arc length parameterization, specified by the text, the inner product is always zero. This is because the derivative of a unitized vector (the tangent vector, itself simply the derivative of position in an arc length parameterization) is always orthogonal to that unitized vector. While the formula as it stands is nicely consistent with the rest of the presentation, there should at least be an observation that this step of the process is a mere formality, and the subtractive correction in this first step is unnecessary. — Preceding unsigned comment added by 206.116.24.195 (talk) 00:47, 9 January 2018 (UTC)

You may be interested in the changes I just made, which make everything in that section independent of parametrization. — Preceding unsigned comment added by 77.2.105.171 (talk) 20:49, 22 January 2019 (UTC)

(Has this been fixed?)

Not to knit pick, but there is a hole in the logic. After the derivation of dB/ds, it is concluded that this is parallel to N, Showing it perpendicular to T, but not B. —The preceding unsigned comment was added by 144.133.97.254 (talk • contribs) 11:06, 20 June 2005.

Since B is defined as T X N, B is perpendicular to both T and N. Hence, if dB/ds is parallel to N, it is perpendicular to everything N is -- in particular, B. —The preceding unsigned comment was added by 205.251.145.10 (talk)

History of use of linear algebra and vector notation in the formulas

The Frenet page notes (correctly) that "[h]e wrote six out of the nine formulas, which at that time were not expressed in vector notation, nor using linear algebra." That is, they were written out explicity using coordinates.

There is a textbook, "A treatise on the analytic geometry of three dimensions," by Salmon which uses coordinates exclusively. This book was first published in 1862, but was revised several times. I've been reading the fifth edition published in 1915. It, too, uses coordinates.

Does anyone know when differential geometry matured beyond three dimensions and the use of coordinates? Presumably it was in the early 20th century. Lunch 02:14, 13 August 2006 (UTC)

OK, so Riemann first described more general geometries in the mid 1850s, and by the end of the century this was well-known. So had the use of vector notation and linear algebra been popularized by then? (But Frenet had only published his coordinate-based formulas in 1847...) Lunch 02:23, 13 August 2006 (UTC)

Intriguing question. In the history of the development of modern coordinate-free concepts of geometry, there are probably too many threads to disentangle. Rather, the use of vector notation (among other things) resulted from a favorable confluence of ideas. Riemann, certainly, was the chief influence. But remember that his thinking was also firmly situated in the realm of coordinates — though the geometrically relevant facts should remain independent of those coordinates in a manner not fully explained until later via Christoffel's covariant symbols. On another front, Peano in 1888 introduced a coordinate-free axiomatic formalism for spaces of vectors (what we now know of as vector spaces) based on the work of Hermann Grassmann (of some forty years prior). As for who wrote down the first recognizably modern version of the Frenet-Serret formulas, my money's on the French (Darboux, maybe Cartan). But I'm only guessing at that. Silly rabbit 00:46, 7 May 2007 (UTC)

Ribbons

Nice to see the ribbons section. There is another version of a ribbon which has slightly different mathematics. Take a strip of paper with a curve draw down the middle of it, define the T to be the tangent to the curve and N to be the normal to the strip (note this is different to the frenet normal), let B be the cross product of T and B. We now have three different quantaties κ, g and τ which describe the behaviour of the strip. $\kappa =T'\cdot N$, $g=T'\cdot B$ and $\tau =N'\cdot B$. Examples of these are: a strip cut round a cylinder has κ=constant, g=0, τ=0, a circle in the plane has κ, g=constant, τ=0 and a twisted ribbon has κ=0, g=0 and τ=constant. Konederink described these in Koenderink J.J.: Solid shape. MIT Press, Cambridge, Massachusetts, London, England, 1990. --Salix alba (talk) 09:50, 21 May 2007 (UTC)

Thanks. That's worth checking out. Silly rabbit 11:13, 21 May 2007 (UTC)

I think that the assertion of the ribbon being developable is false. Consider the case of a circular helix. The surface spanned by the normal vectors is a helicoid, which is a known non-developable surface. However, the surface spanned by the tangent vectors is developable, but it isn't regular along the curve. However, before editing it would be nice to read the mentioned book(s), because the curvature formula may mean something true. Marcosaedro (talk) 23:14, 27 December 2009 (UTC)

I concur with Marcosaedro that the "Frenet ribbon" as defined is not developable in general. Initially I thought that the helicoid counterexample was an anomalous exceptional case, since it is the ruled surface containing the Frenet ribbon of the curve with constant curvature and torsion. But since the Gauss curvature of this surface is non-zero, for nearby generic space curves the corresponding Gauss curvature will still be non-zero, and so the ruled surface is not developable. This furnishes several more counterexamples. In "Lectures on Differential Geometry" (1965), on pages 252,253,254, Sternberg defines ribbons, the Frenet ribbon, and a development operation which assigns to *any* ribbon a *plane curve* and hence a plane curvature (perhaps the identity given in the article is true for this plane curvature). He does not claim that the Frenet ribbon is developable. Jimmymath (talk) 17:00, 6 June 2017 (UTC)

I checked the book by Sternberg too and found exactly what Jimmymath claimed. Thus this section seems like original research (not to mention it is probably wrong). Tony Beta Lambda (talk) 06:59, 24 December 2019 (UTC)

Out of vanity I looked back here to admire my comment, and found that the offending material was never actually changed. I finally removed the poor orphaned curvature formula. A little bit of searching turned up two pages where the exact same material appears. (Due to Wikipedia's spam filter, I am not able to write the URLs here, even as plain text. Anyone know how to include blacklisted URLs not as links but just as a factual record of their spamminess?) In fact, much of the rest of the Wikipedia article on Frenet-Serret appears on those two pages as well. Jimmymath (talk) 00:15, 24 February 2020 (UTC)

Frenet-Serret formulas can be generalized

We can make an other trihedron with angular speed and with normal unit vector. For this, please see my researches. --Abel 05:22, 16 June 2007 (UTC)

Proof?

I challenge the statement : "Note the first row of this equation already holds, by definition of the normal N and curvature κ. So it suffices to show that (dQ/ds)Q^T is a skew-symmetric matrix."

Why does skew-symmetry imply full knowledge of the matrix when the first row of the matrix is already known. Surely, we still don't know anything about the 3-2 entry of the matrix. Randomblue (talk) 14:10, 4 February 2008 (UTC)

We don't need to "know" the 2-3 entry. We'll just label it with the letter τ. (This is, in effect, the definition of torsion.) Then the 3-2 entry is -τ, etc. The proof is fine the way it is. Silly rabbit (talk) 14:23, 4 February 2008 (UTC)

New post

After looking at the article, it seems to list B as || r(t) X r'(t) ||... when it should actually be || r'(t) X r(t) ||. I would fix it myself, but I don't know how to manipulate the formulas. —Preceding unsigned comment added by 132.216.48.150 (talk • contribs)

Now I see what you were referring to. You mean that

$\mathbf {B} (t)=\mathbf {T} (t)\times \mathbf {N} (t)={\frac {\mathbf {r} '(t)\times \mathbf {r} ''(t)}{\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|}}.$

rather than

$\mathbf {B} (t)=\mathbf {T} (t)\times \mathbf {N} (t)={\frac {\mathbf {r} ''(t)\times \mathbf {r} '(t)}{\|\mathbf {r} ''(t)\times \mathbf {r} '(t)\|}}.$

Yes, I think that's right. I will make the change. Thanks, siℓℓy rabbit (talk) 13:26, 5 November 2008 (UTC)

Discussion of removal of proof

I have reverted the removal of the proof. The article has included a proof of this result for several years now. I would like to see a little more discussion of the wholesale removal of the section than the glib edit summary "rm unneeded proof; this is unimportant". Actually, the proof is quite important, for instance for the study of more general moving frames. Otherwise I should think that Phillip Griffiths would not have spent time discussing the proof in a paper in the Duke Mathematical Journal. Clearly this proof is notable, and I see no valid reason removing it. siℓℓy rabbit (talk) 23:41, 13 November 2008 (UTC)

I agree that some mention of the proof is informative. I'm not convinced that it needs a separate section of its own. We could simply state that as Q is orthogonal then its derivative is skew-symmetric (Orthogonal matrix#Lie algebra). Then is a simple case of observing that $\mathbf {T} '\cdot \mathbf {B} =0$ to show it is in the form specified. --Salix (talk): 00:21, 14 November 2008 (UTC)

That seems to be reasonable. However, the old version of the article included two lengthy sections on the proof, which was clearly overkill. The present version is, I hope, a welcome compromise between the two extremes. Anyway, the proof essentially does explain just what you say, but with just a few more details. siℓℓy rabbit (talk) 00:41, 14 November 2008 (UTC)

Frenet-Serret image

Moved from Wikipedia:Village pump (miscellaneous) ([1])

Go ahead, move it to where you see fit. I was just trying not to bloat the article and place it somewhere where it would be clear. About the whitespace, I constantly have those problems (because I export my images from Mupad/Maple) and I havent been able to figure it out yet. If you know a way, then please do trim it (and tell me how). Anyways, I'll be looking into that. Goldencako 23:55, 13 November 2008 (UTC)

Whitespace in formulas

Option 1: Without whitespace	Option 2: With whitespace
${\begin{matrix}{\frac {d\mathbf {T} }{ds}}&=&\kappa \mathbf {N} \\&&\\{\frac {d\mathbf {N} }{ds}}&=&-\kappa \mathbf {T} +\tau \mathbf {B} \\&&\\{\frac {d\mathbf {B} }{ds}}&=&-\tau \mathbf {N} \end{matrix}}$	${\begin{matrix}{\frac {d\mathbf {T} }{ds}}&=&&\kappa \mathbf {N} &\\&&&&\\{\frac {d\mathbf {N} }{ds}}&=&-\kappa \mathbf {T} &&+\,\tau \mathbf {B} \\&&&&\\{\frac {d\mathbf {B} }{ds}}&=&&-\tau \mathbf {N} &\end{matrix}}$

The whitespace is useful in the expression of the Frenet-Serret formulas. It is fairly common to do this with linear systems of this kind, since it organizes the coefficients in the same way in the various rows and gives a visual cue for what goes where. So I prefer it the way it was. Sławomir Biały (talk) 16:06, 4 March 2012 (UTC)

I added at the top of this section the two formats we are discussing about. I prefer the format without added whitespaces, and I have never seen the other format elsewhere. Somebody (e.g. Hanson, A.J. (2007), http://www.cs.indiana.edu/pub/techreports/TR407.pdf {{citation}}: Missing or empty |title= (help)) prefers using the matrix notation, rather than the three formulas:

${\begin{bmatrix}\mathbf {T'} \\\mathbf {N'} \\\mathbf {B'} \end{bmatrix}}={\begin{bmatrix}0&\kappa &0\\-\kappa &0&\tau \\0&-\tau &0\end{bmatrix}}{\begin{bmatrix}\mathbf {T} \\\mathbf {N} \\\mathbf {B} \end{bmatrix}}.$

In this case, the visual cue becomes evident. Perhaps, we should insert the matrix format in the introduction. Paolo.dL (talk) 17:41, 4 March 2012 (UTC)

The white space version is quite familiar to me, I can't remember which book it came from. The non-white space version, seems a bit odd its not left aligned as formula would normally be presented. The matrix version is a bit over the top for the intro. --Salix (talk): 19:47, 4 March 2012 (UTC)

You are right: the centered typographic alignment in option 1 is unusual. There's a another option, the simplest:

Option 3: Left aligned
${\begin{aligned}{\tfrac {d\mathbf {T} }{ds}}&=\kappa \mathbf {N} \\{\tfrac {d\mathbf {N} }{ds}}&=-\kappa \mathbf {T} +\,\tau \mathbf {B} \\{\tfrac {d\mathbf {B} }{ds}}&=-\tau \mathbf {N} \end{aligned}}$

If you feel that option 2 is more commonly used than 3, or even as commonly used as option 3, that would be a good reason to restore opition 2 in the intro. Otherwise, I think we should use option 3. Please feel free to edit accordingly. Paolo.dL (talk) 20:31, 4 March 2012 (UTC)

It seems fair to notice that option 2 was originally introduced in the article by User:Silly rabbit. He is now retired, but he made valuable contributions as a Wikipedia editor and his opinion should be taken into account here. So, I seem to be the only one who does not like option 2. Thank you for sharing your opinions. Paolo.dL (talk) 00:04, 5 March 2012 (UTC)

Formulas in N dimensions

Any chance that a proof of the last two equations in this section could be supplied? I know from other reading that they are correct, but for the life of me I don't see how they follow from the previous material.

I just put another piece of some of my hand written notes online: https://gitlab.com/xaverdh/math/-/jobs/823542122/artifacts/file/math-notes.pdf Chap. 2 contains a proof — Preceding unsigned comment added by 2A01:C22:7247:4900:B82B:5525:EC11:C15E (talk) 15:01, 1 November 2020 (UTC)

Ribbon diagram

Maybe it is my misunderstanding as I am only just learning about the Frenet-Serret frame, but to my mind the the ribbon in the diagram would be formed by sweeping with a line segment [−B,B] as opposed to [−N,N] as stated in the text. Wat7sy (talk) 13:48, 7 October 2016 (UTC)

All Diagrams

I can't see any images in any diagram on this page. Though I have an old 2001 PC with Windows XP, I see every other Wikipedia page clearly, diagrams and all. Please check source code on this. — Preceding unsigned comment added by Tamjk (talk • contribs) 17:05, 29 November 2016 (UTC)

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o(s^2) Vs O(s^3)

The taylor expansion gives residual errors of o(s^3), which is correct, although i think a stronger condition is O(s^4) Aspaine (talk) 06:58, 4 October 2019 (UTC)