Error term in pointwise approximation of the curvature of a curve

Let P be a polygonal line approximating a planar curve Γ, the discrete curvature kd(P) at a vertex P∈P is (usually) defined to be the quotient of the angle between the normals of the two segments with vertex P by the average length of these segments. In this article we give an explicit upper bound of the difference |k(P)−kd(P)| between the curvature k(P) at P of the curve and the discrete curvature in terms of the polygonal line's data, the supremums over Γ of the curvature function k and its derivative k′, and a new geometrical invariant, the return factor ΩΓ. One consequence of this upper bound is that it is not needed to know precisely which curve is passing through the vertices of the polygonal line P to have a pointwise information on its curvature.