Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 25I / a / Solution

The fundamental theorem of regular space curves states that smooth functions $\kappa (s)>0$ and $\tau (s)$ determine a unit-speed regular curve in ${\mathbb{R}}^3$ with that curvature and torsion, uniquely up to a proper Euclidean motion of Euclidean three-space. Existence follows by solving the Frenet-Serret formulas for an oriented orthonormal frame and then integrating its tangent vector.