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Abstract

Accurate estimations of geometric properties of a smooth curve from its
discrete approximation are important for many computer graphics and computer
vision applications. To assess and improve the quality of such an
approximation, we assume that the curve is known in general form. Then we can
represent the curve by a Taylor series expansion and compare its geometric
properties with the corresponding discrete approximations. In turn we can
either prove convergence of these approximations towards the true properties as
the edge lengths tend to zero, or we can get hints on how to eliminate the
error. In this paper, we propose and study discrete schemes for estimating
tangent and normal vectors as well as for estimating curvature and torsion of a
smooth 3D curve approximated by a polyline. Thereby we make some interesting
findings about connections between (smooth) classical curves and certain
estimation schemes for polylines.