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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?

George
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1 Answers1

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

Robert Israel
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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? – George Dec 26 '12 at 08:59
  • 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. – Robert Israel Dec 26 '12 at 09:48