\(L_\infty\)-locality of three-dimensional Peano curves (Q2280444)

The authors investigate the \(L^\infty\)-locality of three-dimensional Peano curves \(p: I\to I^3\) where \(I\) denotes a closed interval. Here, locality means that points on the curve are close to points in \(\mathbb{R}^3\). As a quantitative measure of locality the upper bound of the expression \[ \frac{\|p(t_2) - p(t_1)\|^3}{t_2-t_1} \] is used, where the underlying metric is the \(L^\infty\)-metric. Algorithms for the fast and exact computation of the above ratio are presented.