Universal method for solving curves when given curvature and torsion

Question

For planar case, given curvature $\kappa$, we can take $x'(s)=cos \theta(s), y'(s)=sin \theta(s)$ and get $\theta(s)=\int \kappa(s)ds$, then solve two "independent" ODE (that is, only an integral) to get the curve.

However, for spatial case, I don't know whether there is a universal method to solve curves when given curvature and torsion.

Any help will be appreciated.

Answer 1

A general space curve is given by a fourth order differential equation in three dimensions:

$$\boldsymbol{x}^{(4)}- \left( \frac{2\kappa'}{\kappa}+\frac{\tau'}{\tau} \right) \boldsymbol{x}'''+ \left( \kappa^{2}+\tau^{2}+ \frac{2\kappa'^{2}-\kappa \kappa''}{\kappa^{2}}+ \frac{\kappa' \tau'}{\kappa \tau} \right) \boldsymbol{x}''+ \kappa^{2}\left( \frac{\kappa'}{\kappa}-\frac{\tau'}{\tau} \right) \boldsymbol{x}'= \mathbf{0}$$

which is only solvable for some special cases.