 Fundamental theorem of regular space curves

Smooth functions $\kappa (s)>0$ and $\tau (s)$ determine a unit-speed space curve with curvature $\kappa$ and torsion $\tau$, uniquely up to a proper Euclidean motion of Euclidean three-space. Solve the Frenet-Serret formulas for an oriented orthonormal frame and integrate its tangent for existence; uniqueness for the frame ordinary differential equation gives rigidity.

This is the space-curve version of the fundamental theorem of curves.