Discrete curvature and torsion-based parameterization scheme for data points

Abstract

Parameterization for data points is a fundamental problem in the field of Computer Aided Geometric Design. Recently, an improved centripetal parameterization technique is proposed by Fang et al and its superiority to other common methods, such as uniform method, chord length method, centripetal method, Foley method and universal method, etc., is validated by a lot of numerical examples. In this article, a refined formula is firstly given to resolve the problem that the parameter values obtained by Fang's method are usually a little bigger than the optimum ones. Furthermore, as enlightened by the local differential geometric properties of parametric curves, a novel parameterization scheme is put forward by introducing the discrete curvature and torsion information at each point. As indicated by numerical experiments, the deviation measured by a curvature and Euler distance-based criterion between the interpolation B-spline curve obtained by our method and the polyline constructed by the points are smaller than the ones between the curves obtained by Fang's method, the chord length method, the standard centripetal method and the polyline. The proposed algorithm is applicable to both 2D and 3D points and has a distinct advantage for 3D points as compared with the three aforementioned methods.