# Can I solve the Frenet-Serret formulas with the only assumption that the cirvature-torsion of the curve are constant?

I am trying to find the general equation for space curves which have constant curvatures throughout their length. In general I am interested for curves of more than 3 dimensions.

Assuming that all curvatures are constant for the entire length of the space curve, can I use the frenet serret formulae to derive the most general representation of such a curve?

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are there Frenet Serret formulas for more then three dimensions? The formulas I know are very much three-dimensional in nature. I'm interested to see what the community has to offer here. –  James S. Cook Dec 24 '12 at 19:28
@James: Wikipedia credits Jordan with the higher-dimensional generalization of the Frenet-Serret formulas. –  Rahul Feb 28 '13 at 21:38
@RahulNarain neat, I had not seen those before. Should make nice homework for some course. –  James S. Cook Mar 1 '13 at 13:58

I assume the curvature $\kappa$ is not $0$. For one of your curves, the tangent vector $T$ moves on the unit sphere in an arbitrary way, constrained only by $$\left| \dfrac{dT}{ds} \right| = \kappa$$ Thus its path can be any $C^1$ curve on the sphere, which you traverse at constant speed $\kappa$ (with respect to the parameter $s$). To get the actual curve in space, you then integrate: $$X(s) = \int_0^s T(t)\ dt$$