Research On Design Of Developable Surfaces And Ball Bézier Curves In CAGD

In this paper, some researches have been done on design of developable Bezier surfaces and Poisson surfaces, the boundary surfaces of ball Bezier curve in Computer Aided Geometric Design (CAGD).In Chapter 1, an overview of development of Bezier curves and surfaces is presented, and hence developable surfaces and ball Bezier curve are introduced.In Chapter 2, an algorithm is presented that generates a developable Bezier surface through a Bezier curve called a directrix. The algorithm is based upon the differential geometry theory about the necessary and sufficient conditions for a surface which is developable, the degree elevation formula for parameter curves and the linear independency for Bernstein basis. No nonlinear characteristic equations have to be solved. Moreover the vertex for a cone and the edge of regression for a tangent surface can be given easily.In Chapter 3, design of developable Poisson surface is presented. Based on the necessary and sufficient conditions for a surface which is developable, the degree evaluation formula for Poisson curves and the linear independency for Poisson basis functions, an algorithm that generates developable Poisson surfaces through a Poisson curve of arbitrary degree and shape is given. These results could be applied to design of surfaces frequently met in revolution cutting, and plate expansion of spiral-like pipe surfaces.In Chapter 4, how to solve boundary surface of a ball Bezier curve is presented. With the envelope algorithm of the family of space surfaces in differential geometric and variable transformation, an accurate representation for the boundary of a ball Bezier curve was gained;and furthermore, it was approximately represented as a Bezier surface or an union of Bezier patches by using Legendre polynomial best square uniform approximation. Ball Bezier is a kind of error control and error analysis with simple expression, less storage and fast computation.