# How can I prove that a curve with constant nonzero curvature and torsion is a helix? [duplicate]

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.