4
$\begingroup$

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?

  • 0
    @RahulNarain neat, I had not seen those before. Should make nice homework for some course.2013-03-01

1 Answers 1

1

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$

  • 0
    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