Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 25H / c / Solution

The fundamental theorem of regular space curves states that smooth functions $k>0$ and $\tau$ on an interval determine a smooth arc-length-parametrized curve in ${\mathbb{R}}^3$, unique up to a proper Euclidean motion. Consequently two such curves $\alpha$ and $\tilde{\alpha }$ are related parameterwise by a proper Euclidean motion if and only iffor every $s$.