Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 26G / c

Write a proper Euclidean motion of Euclidean three-space as $E(x)=Ax+b$, where $A\in SO(3)$. ThenBecause $A$ preserves norms, dot products, and cross products, the formulasshow directly that curvature and torsion are unchanged by $E$.

For the shifted curve ${\gamma }_{s_0}(s)=\gamma (s-s_0)$, every derivative at $s$ equals the corresponding derivative of $\gamma$ at $s-s_0$. ConsequentlyBoth are therefore pointwise Euclidean invariants in the stated sense.