General Bézier Curves With Adjustable Free Parameters

The Bezier Model is a basic designed scheme for the parameter curves and surfaces in CAGD, which is based on Bernstein basis. When the control vertex and its corresponding Bernstein basis are given, the Bezier curves or surfaces are confirmed. A class of extential Bernstein basis function with shape parameters is presented in this thesis. By varying the values of the shape parameter and no changing the control vertex, the shape of the general Bezier curves or surfaces based on extential Bernstein basis function can be adjusted. This thesis focuses on the problems as follows:Firstly, the research background is introduced in the chapter one.Secondly, a class of extential Bernstein basis function with shape parameters is presented in the chapter two, which is an extension of the Bernstein basis function and has positivity, weighting property and symmetry, when the parameters are in their numeric area. The new basis possesses elegant representations and holds some remarkable properties similar to that of the Bernstein basis.Thirdly, with the extential Bernstein basis function in the chapter two, a class of general Bezier curve with adjustable free parameters is defined in the chapter three. The properties of the general Bezier curves have similar with the Bezier curves, such as endpoints' properties, symmetry, convex hull, variation decrement (V.D.) and geometric invariability.Fourthly, some applications of extential Bernstein basis function with shape parameters are given in the chapter four.Finally, in the chapter five this thesis draws the conclusions and talks about some prospects.