Curves with Constant Curvature

Question

Besides lines, circles and helices, are there any other curves that have a constant curvature $\kappa$?

If there exist any, are there any that have an explicit formula in the form of r(t) = [x(t) , y(t) , z (t)] ?

Answer 1

Yes, of course. You've just listed those with constant curvature and torsion.

I like to think of curves of constant curvature in terms of the Darboux vector $$\omega = \tau T + \kappa B$$ which gives the axis-angle of infinitesimal rotation of the curve's frame at every point. Here $\tau$ is free, so you can get different curves of constant curvature by changing how much you twist about the tangent as you travel along the curve.

You can find formulas for the curves $r(t)$ by solving the Frenet-Serret equations: \begin{align*} r'(t) &= T(t)\\ T'(t) &= \omega \times T(t)\\ N'(t) &= \omega \times N(t)\\ B'(t) &= \omega \times B(t) \end{align*} which is a system of first-order ODEs that can be integrated up given appropriate initial values. Whether or not $r$ has a closed form will depend on the complexity of your chosen torsion $\tau(t)$.