Talk:Frenet–Serret formulas
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[edit] (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)
[edit] 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)
[edit] 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. 
, 
and 
. 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)
[edit] 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)
[edit] 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)QT 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)
