A novel extension of the Bézier model and its applications to surface modeling

The construction of Bézier curves and surfaces with shape parameters is one of the most popular areas of research in CAD. In this paper, we present a novel shape-adjustable generalized Bézier curve with multiple shape parameters and discuss its applications to surface modeling in engineering. Firstly, a class of new generalized Bernstein basis functions with explicit expressions is proposed, which is a natural extension of the classical Bernstein basis functions of degree n. Furthermore, the corresponding Bézier curves and surfaces with global and local shape parameters are constructed and their properties are also studied. The shapes of the curves and surfaces can be adjusted intuitively and foreseeably by changing the shape parameters. Secondly, in order to tackle the problem that most complex curves in engineering often cannot be constructed by using a single curve, we derive the necessary and sufficient conditions for C1 and C2 continuity of these new curves. Finally, using shape-adjustable generalized Bézier curves, we construct six different types of engineering surfaces with multiple shape parameters, including general cylinders, bilinear surfaces, ruled surfaces, swung surfaces, swept surfaces and rotation surfaces. The modeling examples show that the proposed methods are effective in geometric modeling.