The Fairing Of The Fourth Bézier Curves

The fairing of curves and surfaces has been widely studied in the field of CAGD due to its great theoretical and practical value. In the mechanical parts design, parametric curves and surfaces are usually applied to interpolate, approximate and fit the giving data. Nevertheless, errors inevitably exist in measuring and calculating data, it is necessary to fair the curves and surfaces obtained from these data for a desiring curve and surface.The faring of curves can be achieved by revising the data points or the control points. Thus the problems of fairing curves are researched from two aspects, i.e. fairing through revising the data points and revising the control points in this paper. In practice, the 'bad' point may be single or continuous, so the cases of modifying a single point or modifying more than one point in each modification step are discussed respectively. The validity of various fairing algorithms are analyzed through the energy and the energy with error-controlling complements, especially two fairing methods are given and the efficiency of the fairing methods are compared in case of modifying two points simultaneously. Experimental tests shows that the fairing algorithms of this paper have their advantages, thus we have choices between modifying one single point or modifying more than one point simultaneously, even choices between different fairing methods of modifying more than one point simultaneously in practice, so to improve the efficiency of the fairing processes. In this paper we use common fourth Bézier curves as fairing subjects. Its structure and the main results are as follow:In the first part, the significance and the research status of the fairing are discussed.In the second part, the concepts and properties of the fourth Bézier curves are reviewed and the inverse calculation of control points of the fourth Bézier curves is introduced.In the third and the fourth part, the problem of global and local fairing of the fourth Bézier curves is researched systematically from two aspects, i.e. the modifying of the control points and the data point respectively. Firstly, the algorithms and the procedure of the algorithms of modifying one point in a step are given with some fairing examples. Secondly, the fairing algorithms of modifying two points simultaneously are obtained, the properties of the algorithms are also analyzed and their efficiencies are compared. Finally, the algorithm of fairing there points simultaneously is put forward. The validity of the proposed methods is demonstrated via experimental tests.In the fifth part, the conclusions of this paper are summarized and the future work is put forward.The fairing of the fourth Bézier curves is obtained by energy minimization. The methods can be applied for both global and local curves fairing.