Research On Fairing Of Curves And Surfaces With Target Curvatures

Designing of curves and surfaces to meet user’s requirements is a basic problem incomputer aided design. There are many respects of requirements on curves and surfaces indesign of products, in which fairness is one of the most important. If the curve or surface isnot fair, it will not meet requirements in functional and aesthetic aspects and be difficult tomanufacture. Therefore, in a long period fairing of curves and surfaces is a hot topic in thefield of CAD, and it receives great attention in both industry and academia area. Althoughup to now many fairing methods have been proposed, but, because of the complexity of thefairness, so far it is still an open problem. In this paper, after depth analysis of the meritsand demerits of existed fairing methods, new fairing methods for curves and surfaces basedon target curvatures are presented.Our main contributions are summarized as follows:1. A new algorithm for curve fairing based on target curvatures is presented. First, thecurvatures of the original curve on some sample points are computed, then, targetcurvatures of faired curve are obtained by smoothing the original curvatures with digitalfiltering methods, finally, the fairing problem is changed to an optimization problem, that is,the squared sum of the deviations from the curvatures of faired curve and the targetcurvatures on the sample points is minimized subject to the movement from the originalcontrol vertices to the faired ones is within the given tolerance. Considering it is difficultto handle with the traditional numerical optimization methods, genetic algorithm is used tosolve this optimization problem. The experimental results show that our algorithm for curvefairing has better performance than some widely used methods.2．Algorithms for surface fairing based on target curvatures and second derivatives arepresented. The basic idea is: first, the original principal curvatures of the surface on somerectangular grid points are computed, then, target principal curvatures of the surface areobtained by smoothing the original principal curvatures with db4wavelet transformation,finally, the control vertices of the original surface are modified within the given tolerancesuch that the squared sum of the differences between the principal curvatures of the originalsurface and those of the faired surface on the given points is minimized. Because thecomputing of the principal curvatures is very time-consuming, we replace the principalcurvatures with the second partial derivatives of the surface with respect to the parameters uand v and change the nonlinear constraints into linear constraints approximately in practicalcomputation, finally the fairing problem is converted into a quadratic programmingproblem and it can be solved efficiently. Experimental results show that our surface fairingmethod has good performance.