B-spline Curve Smoothing

Free-form curves and surfaces are widely used in shape design and reverse engineering. In CAD system, parametric curves and surfaces are usually applied to interpolate, approximate and fit the giving data. Nevertheless, As error exists while measuring and calculating data, it is necessary to do fairing operations on the curves and surfaces obtained by these data, thus to get the aesthetic curves and surfaces. Fairing curves and surfaces is becoming one of the most important problems in the research of computer-aided design.B-spline curves or surfaces are commonly-used tools for shape design and data fitting. The problem of how to fair cubic planar B-spline curves is discussed in this paper. The shape of a B-spline curve is determined by its control polygon. Therefore one can get the fair curve by vibrating its control points. In this thesis, fairing algorithms based on energy minimization are presented. First, we present an interactive algorithm about the fairness of a B-spline curve. The basic idea is to give each control point a small perturbation. A weight of each control point is also applied to adjust the perturbation together with the energy function. Thus, the purpose of faring can be achieved by confining the shape of the curve. Though the perturbation of the B-spline curve can be reflected by the perturbation of its control points, they are not precisely the same. So another constrained faring algorithm is given in this thesis. Suppose the fair curve is approaching to some given data. We put constrains on energy functions and finally the fairing curve can be obtained by solving a linear system of equations.The fair B-spline curve is obtained by energy minimization. The method can be applied for both global and local curve fairing. Examples of B-spline curve fairing from simulated and real data are presented to show the efficiency of the method. The whole process of faring need no recursive computations or optimizations. As a result, it is easy to carry out. In addition, users can get fair curves in different settings by adjusting the number of the constrained points, the locations as well as their weights.