On Shape Modification Of NURBS Curves And Surfaces

Shape modification of NURBS curves and surfaces is one of the most crucial techniques in CAGD. Many researchers have worked actively to develop more convenient and effective methods in this area. However, since the practical applications always vary and the requirement for convenience of operation becomes more and more important, a lot of problems remain to be solved. This thesis presents some effective methods for shape modification of NURBS curves and surfaces. The main results obtained are as follows:Firstly, based on the analysis of the foregone modification methods, geometric features modification of NURBS curves via constrained optimization is presented. Given a NURBS curve, perturbation for control points is chosen and the LSM (Least Square Method) is used to modify the geometric features such as position, tangency or curvature of the selected parametric point on the given curve. It is also shown that the proposed method can be extended to the case for modifying geometric features at a series of selected points. Moreover, based on energy optimization, another approach to geometric features modification is presented. The optimization objective is to minimize the strain energy of error curve. The cases for modifying geometric features such as position, tangency or curvature of one selected point and a series of selected points are discussed respectively. The numerical examples show that the former method is very convenient and interactive whereas the latter yields that the modified curve tends to be linear.Secondly, for geometric features modification of NURBS surfaces, a method via constrained optimization is given to modify the geometric features such as position, tangency or normal vector of the selected parametric point on the given NURBSsurface by minimizing the changes of control net by means of the LSM. Furthermore, it is shown that the proposed methods can also be extended to the case for modifying geometric features at a series of selected points. Explicit formulae are also obtained to compute new control points of the modified surfaces. On the other hand, by means of energy optimization, another approach to geometric features modification is presented. The optimization objective is to minimize the thin plate energy of error surface. Also the cases for modifying geometric features such as position, tangency or normal vector of one selected point and a series of selected points are discussed respectively. And the numerical examples show that the energy optimization method gives natural effects at the cost of high computation.Thirdly, a method for shape modification of NURBS surfaces based on the idea of constructing Coons surfaces is presented. The modified surface satisfies specified geometric constraints at a series of selected parametric points. For the given NURBS surface, a local surface satisfying geometric constraints is constructed on each parametric point. And a total surface is constructed by means of these local surfaces. Then a redundant surface satisfying the redundant constraints is interpolated. At last, applying the idea of constructing Coons surfaces, the modified surface is obtained.