Talk:Torsion of a curve

A curve is planar if and only if it's torsion is 0 and it is twice differentiable, I believe. I'm not sure who watches this page, but that ought to be changed.

"Torsion is analogous to curvature in two dimensions."

Dunno that I like that analogy. Curves in 2D have zero torsion (always). Maybe a better analogy might involve a corkscrew (on the physical side) or a helix (on the abstract side).

This page and the one on curvature might stand to see more discussion of the different formulas for torsion and curvature of parametric curves. There should also be a link to the Frenet-Serret formulas. (That page - on the Frenet formulas - looks nice.)

Lunch 02:06, 28 February 2006 (UTC)

Torsion and curvature are two separate quantities that together define a curve in three dimensions. It is even sometimes called "Second Curvature" because of this. The two quantities define different properties: curvature is the magnitude of the curvature vector, while torsion is the ratio of the normal vector and the derivitive of the binormal vector. They share the properties of defining how a curve is shaped, and that they are intrinsic properties of the curve, but the relationship between them is definitely not just an analogy.

Do we need this as a separate article?

I think this article may be a good beginning section of Torsion tensor, which deals with the general usage of torsion in differential geometry. (Torsion (differential geometry) redirects there.) There is too little to say only about 1-dimensional cases; that is why we don't have Curvature of curves article. How do you think? --Acepectif 06:35, 16 November 2007 (UTC)

No. Besides the somewhat relevant fact that the torsion of a curve is not an example of torsion in the sense of the torsion tensor, the other article is essentially impenetrable. What this article needs is examples and figures. If there is any article it can be merged with, that would be Frenet-Serret formulas.Arcfrk (talk) 04:49, 30 January 2008 (UTC)

Mathematical error?

Maybe i'm just unable to calculate it correct, but it seems that there is a mistake in the formulas for torsion. The first formula has the denominator ${\displaystyle {\left\|{r'\times r''}\right\|^{2}}}$, which is equal to ${\displaystyle \left({r'\times r''},{r'\times r''}\right)}$ = ${\displaystyle \left(\left(y'z''-z'y'',z'x''-x'z'',x'y''-y'x''\right),\left(y'z''-z'y'',z'x''-x'z'',x'y''-y'x''\right)\right)}$ = ${\displaystyle \left(y'z''-z'y''\right)^{2}+\left(z'x''-x'z''\right)^{2}+\left(x'y''-y'x''\right)^{2}}$, which is obviously not equal to denominator of the second formula ${\displaystyle (x'^{2}+y'^{2}+z'^{2})(x''^{2}+y''^{2}+z''^{2})}$. Penartur (talk) 22:46, 19 June 2011 (UTC)

I totally agree with this remark. The formula for the torsiune is wrong. As Penartur noticed, the denominator of the last expression is wrong.

— Preceding unsigned comment added by Nico2011 (talkcontribs) 21:10, 2 January 2012 (UTC)

Animation not animated

I cannot see the animation moving. Is there something wrong with it or it is just me? I'm using firefox in a Debian wheezy box. Juliusllb 09:56, 20 November 2013 (UTC) — Preceding unsigned comment added by Juliusllb (talkcontribs)

For me it only plays when viewed at full size. — NuclearDuckie (talk) 14:45, 9 April 2014 (UTC)
Same for me. —DIV (120.17.85.139 (talk) 04:39, 22 June 2017 (UTC))

"any space curve with constant non-zero curvature and constant torsion is a helix"

Doesn't a helix require non-zero torsion as well, given that constant curvature with zero torsion is a plane curve?

You are right. I have fixed this. (A curve with constant curvature and zero torsion is a circle.) Arcfrk (talk) 08:09, 13 October 2015 (UTC)

Examples / illustrations

Besides the animation, I think it would be instructive to have static images with very simple curves rendered in colour with respectively their curvature and torsion (i.e. images would have to be paired). A suggestion would be a plane ellipse that is then shown projected/bent onto increasingly curved 'planes'. —DIV (120.17.85.139 (talk) 04:42, 22 June 2017 (UTC))