Total Absolute Curvature Estimation

Total (absolute) curvature is defined for any curve in a metric space. Its properties, finiteness, local boundedness, Lipschitz continuity, depending whether there are satisfied or not, permit a classification of curves alternative to the classical regularity classes. In this paper, we are mainly interested in the total curvature estimation. Under the sole assumption of curve simpleness, we prove the convergence, as $ϵ\to 0$ $\epsilon \to 0$ , of the naive turn estimators which are families of polygonal lines whose vertices are at distance at most $ϵ$ $\epsilon $ from the curve and whose edges are in $\mathrm{\Omega }({ϵ}^{\alpha })\cap \text{O}({ϵ}^{\beta })$ $\Omega (\epsilon ^{\alpha })\cap \text{O}(\epsilon ^{\beta })$ with $0<\beta \le \alpha <\frac{1}{2}$ $0<\beta \le \alpha <\frac{1}{2}$ . Besides, we give lower bounds of the speed of convergence under an additional assumption that can be summarized as being “convex-or-Lipschitz”.