Fundamental theorem of regular space curves
= Fundamental theorem of regular space curves
{wiki=Fundamental_theorem_of_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.