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Our class is using Do Carmo's Differential Geometry and I noticed a convention that Do Carmo seems to break when dealing with torsion.

The background is 1.5.1 (what I was working on):

Given $\alpha(s) = (a$ cos$(\frac{s}{c}), a $ sin$ (\frac{s}{c}), b\frac{s}{c})$ where $c^2 = a^2 + b^2$,

b) Calculate $\kappa$ and $\tau$

<p>c) Find the osculating plane of $\alpha$</p>

I found $\kappa$ to be $\frac{a}{c^2}$ and $\tau$ to be $-\frac{b}{c^2}$, but this is where I run into a snag. Do Carmo defines $\tau$ to be $\bar{b}' \bullet \bar{n}$ whereas many other authors (Do Carmo admits this as well) use the convention $\tau = -\bar{b}' \bullet \bar{n}$ instead.

My question is, how does this affect everything else? Is anything changed that I should watch out for? I did notice that now, in the Frenet equations:

$n' = -\kappa t + \tau b$

Note - I'm not asking for help with the aforementioned problem.

Lost
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  • Any formula that uses $\tau$ in one convention must have the $\tau$s replaced by $-\tau$s to obtain the formula for the other convention. That's pretty much the extent of it. – anon Jan 30 '14 at 01:52
  • @anon Alright, thanks. So, for the other convention, I'd get $\tau = \frac{b}{c^2}$ but there would be minimal difference, right? – Lost Jan 30 '14 at 01:55
  • @Lost : I think someone decided that a right-handed helix ought to have positive torsion, and that is the reason for the convention with the minus sign. Does anyone know for sure? – Stefan Smith Jan 30 '14 at 02:11

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