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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.2012-12-24
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    @James: Wikipedia credits Jordan with [the higher-dimensional generalization of the Frenet-Serret formulas](https://en.wikipedia.org/wiki/Frenet%E2%80%93Serret_formulas#Formulas_in_n_dimensions).2013-02-28
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    @RahulNarain neat, I had not seen those before. Should make nice homework for some course.2013-03-01

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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: $$ X(s) = \int_0^s T(t)\ dt$$

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    Thanks for the answer. I was wondering though, is any curve on the sphere a valid curve? Arent only circles the curves with constant curvature?2012-12-26
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    The curve that the tangent vector traces out on the sphere does not have constant curvature. It is just a curve that can be traversed at constant velocity. The space curve with constant curvature is obtained by integration.2012-12-26