A Research On Shape Modification And Singularity Of Bézier Curve And Surface

Computer Aided Geometric Design mainly researches on free curve and surface. And Bezier curve and surfaces are one kind of the most commonly used parametric curves in CAGD and Computer Graphics. It is an important problem to modify the shape of the Bezier curve and surface. Developing more convenient techniques for designing and modifying Bezier curve/surface is an important problem. When the Bezier curve/surface is generated, we always want to modify the shape of the curve/surface in order to satisfy the design. The solution of the practical problem often leads to solving optimization problem.The paper first considers about the Shape Modification of Bezier Curves by Constrained Optimization. This problem is come down to minimization problem of Constrained Optimization. Practical examples are also given, and it also discussed about the single point and multiple points situation.If there are only four control points, the situation is very different. To satisfy the added tangent constrain at end point, the target point can not be located in anywhere. At the end of each chapter, the numerical experiments are given. It sufficiently illuminates the efficiency of the method.At the same time we discussed the Singularities of the Bezier curves, and it also shows how to construct singularities in the reverse problem of Bezier curve. We give the vector of the control points, and it can be expressed by matrix, which makes it possible that we can use it to express Mathematical Models.