Pointwise Euclidean invariant of a curve

A scalar assignment $Q(\gamma ,s)$ is a pointwise Euclidean invariant in the weak covariance sense when it is unchanged by proper Euclidean motions and satisfiesunder translation of the arc-length parameter.

For sufficiently smooth curves, $Q(\gamma ,s)={\kappa }^{\prime }(s)$ 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.