Bézier Curves And Surfaces With Multiple Shape Parameters

Bezier curves and surfaces are one of the important tools for designing curves and surfaces. However, for a given control polygon, Bezier curve's location is determined. To adjust the shape of the curves, one needs to adjust the control polygon, which is not flexible in practical applications. In this thesis, the method introducing parameters in the basis functions is used, by changing the value of the parameter to adjust the shape of the curves.The polynomial basis functions with shape parameters are introduced in this thesis firstly, and the properties of this basis functions are analyzed. Then, the polynomial curves with parameters are defined based on them, the properties of the corresponding curves are analyzed, and the geometric meaning of the shape parameters are studied. Examples show that the defined curves not only inherit the properties of Bezier curves, but also are adjustable in the shape and fit closely to the control polygon. Lastly, Tensor product method is used to extend the curves to surfaces. The surfaces with multiple shape parameters are Constructed, and the properties of the surfaces are analyzed. The graphics display the adjustive effects of the parameters to the surfaces.The study of this thesis shows that the Bezier curves and surfaces with shape parameters provide an effective method for designing curves and surfaces.