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I've gotten as far as $\mathbf{n''}+(\kappa^2+\tau^2)\mathbf{n} = 0$, which suggests or at least permits trigonometric expressions for every component of $\mathbf{n}$ -- not something that seems to lead to a standard expression for a helix. Am I just supposed to prove that a helix has constant nonzero curvature and torsion and invoke the fundamental theorem of curves?

ETA: I was able to get further than the point I described, but the equation for the curve I came up with was...

$\big(c_1\cos{\sqrt{\kappa^2+\tau^2}t}+c_2\sin{\sqrt{\kappa^2+\tau^2}t+c_3t+c_4},c_5\cos{\sqrt{\kappa^2+\tau^2}t}+c_6\sin{\sqrt{\kappa^2+\tau^2}t+c_7t+c_8},c_9\cos{\sqrt{\kappa^2+\tau^2}t}+c_{10}\sin{\sqrt{\kappa^2+\tau^2}t+c_{11}t+c_{12}}\big)$

...and I have no idea how to prove that that's a helix, plus I suspect it isn't even right.

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You can use the Frenet-Serret formulas to find the parametric equations of the curve with constant nonzero curvature and torsion. This gives you a system of three linear differential equations. It's straightforward (albeit a bit tedious) to solve.

However, if you have the fundamental theorem of curves established, all what you need is to show that for every constant nonzero curvature and torsion, there is a helix that has such values. From there, you know that all other curves with the same curvature and torsion are the same helix (up to an isometry).

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    let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/5844/discussion-between-shay-guy-and-ayman-hourieh)2012-09-16
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Take a look at section 1.18 on page 2/5 of this PDF.