Pointwise Euclidean invariants need not depend only on current curvature and torsion
= Pointwise Euclidean invariants need not depend only on current curvature and torsion
For sufficiently smooth curves, $Q(\gamma,s)=\kappa'(s)$ obeys Euclidean and parameter-translation covariance but is not determined by the pair $(\kappa(s),\tau(s))$. Under the bare covariance definition, nonlocal examples such as $Q(\gamma,s)=\kappa(s+1)$ work as well.