What are the conditions for torsion to be zero other than having a plane curve? The only thing I can thing of is an equation that have the torsion that cancels out each other.
Conditions that torsion is zero in a space curve
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2Notice that torsion is usually only defined for curves at points where curvature does not vanish, so to make sense of this some assumption in that direction is needed. – 2010-10-18
3 Answers
As has been said, the curve is planar iff the torsion is zero. This is clear if you look at the Frenet formulas. For fun, you can get from those a formula that the curve $\alpha: \mathbb{R} \rightarrow \mathbb{R}^3$ must satisfy. We need $ \frac{d\mathbf{N}}{ds} + \kappa\mathbf{T} = 0. $ Assuming $\alpha$ is parameterized by arc length, this could be written $ \frac{d}{ds}\left(\frac{1}{\kappa}\frac{d^2\alpha}{ds^2}\right) + \kappa\frac{d\alpha}{ds} = 0 $
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1@vener: Remember that intuitively, torsion measures how much your space curve departs from planarity. It stands to reason that zero torsion gets you a flat curve, as removing the appropriate terms in the Frenet-Serret formulae show. – 2010-10-18
Here's an example of an infinitely differentiable regular curve with identically zero torsion but not contained in any plane. (By regular, I mean that the velocity never vanishes.) \alpha(t) = \left\{ \begin{aligned} &(t,e^{-1/t},0), & t>0,\\ &(0,0,0), &t=0,\\ &(t,0,e^{1/t}), &t<0. \end{aligned} \right.
EDIT: As Mariano Suárez-Alvarez points out in his comment, the torsion is only defined at points where the curvature is nonzero. Since the curvature of this curve is zero at the origin, the torsion is not defined there.
Thus the argument given by yasmar shows that if the curve is regular, its curvature is nowhere zero, and its torsion is everywhere zero, then it's a plane curve.
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0@Mariano: Thanks. That was what I originally thought. But this topic gave me doubts, so I asked for clarifications. – 2010-10-19
Wikipedia states that "if the torsion of a regular curve is identically zero then this curve belongs to a fixed plane." By "regular curve" I expect they mean that the curve's first and second derivatives are never zero. I imagine that a planar curve connected via a linear segment to another planar curve lying in a different plane would still have zero torsion everywhere.
This is the limit of my knowledge; I'm posting it as an answer so others can see it and point out if there are any errors.
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0@Jack Lee @Rahul I was assuming regular was as Rahul defined here, i.e., the curvature doesn't vanish. It sounds like this may not be a universal definition? It does agree with the definition given on this wikipedia page: http://en.wikipedia.org/wiki/Differential_geometry_of_curves I don't have DoCarmo's book on curves and surfaces here, but I would take that as a good authority on what the 'standard' definition should be. – 2010-10-19