带形状参数的Bernstein-Bézier曲面

虽然三角域上的曲面造型方法能有效解决不规则产品的几何造型问题, 在实际工程中有着广泛的应用, 但由于其结构的特殊性和复杂性, 目前对三角域曲面的扩展研究并不多。为了丰富三角域曲面的理论, 针对如何增强三角域曲面形状表示的灵活性进行了专门的研究。首先构造了一组三角域上含一个参数的四次多项式基函数, 它是三角域上二次Bernstein基函数的扩展。然后用递推的方式定义了三角域上含一个参数的n+2次多项式基函数, 它是三角域上n次Bernstein基函数的扩展。基于新的n+2次多项式基函数, 定义了相应的n阶三角域曲面。分析了基函数和曲面的性质, 新曲面不仅具备三角域上Bernstein Bézier曲面的基本性质, 而且还可以在不改变控制顶点的情况下, 通过改变参数的值来自由调整曲面的形状。

Bernstein-Bézier surface with shape parameters

Because the surface modeling method over the triangular domain can effectively solve the geometric modeling problem of irregular products, it is widely used in practical engineering. However, due to the particularity and complexity of the structure, there are not many studies on the extension of triangular surface at present. In order to enrich the theory of triangular surface, the paper carries out a specialized research on how to enhance the flexibility of shape representation of triangular surface. Firstly, a set of polynomial basis functions of degree four with one parameter over the triangular domain is constructed, which is an extension of the quadratic Bernstein basis function over the triangular domain. Secondly, based on it, the basis functions of degree n+2 with one parameter is defined by a recursive way, which is an extension of the Bernstein basis function of degree n. Thirdly, based on the new basis function of degree n+2, the triangular surface of order n is defined. The properties of the basis functions and the surfaces are analyzed. The new surfaces not only have the basic properties of the Bernstein Bézier surfaces, but also enjoy shape adjustable property.