一类有理曲线——RB曲线

A Kind of Rational Curve--RB Curve

为了进一步丰富 Bézier曲线理论 ,首先从 Bernstein基函数出发 ,构造了一类新型函数—— Bernstein函数类 ,同时讨论了它的性质 ;然后用该类函数给出了 Bézier曲线类的生成方法 ;重点研究了一类基于有理形式调配函数的实用曲线—— RB曲线 ,结果表明 ,附加权因子的 RB曲线能部分克服常用的有理 Bézier曲线的权因子的选取没有统一的规则可以遵循的局限 ,提高了曲线设计的灵活性 ;最后给出了实例 ,并得到了可视化结果 .

It is well known that Bézier curve and rational Bézier curve play a very important role in the field of Computer Aided Geometric Design & Computer Graphics. They can be used to generate different curves in given condition. So far, some problems can't be well solved in the regulation of weight factor selecting and the algorithm complexity of curve design. Based on Bernstein polynomials, a new kind of function-Bernstein Function Class is constructed in this paper. Its correlative properties are deduced in details. At the same time, the generating method of Bézier curve class is given. It can be concluded that Bernstein polynomials is a special member of Bernstein function class and the same relation exists between Bézier curve and Bézier curve class. In this article, RB function is chiefly introduced and the method of curve representation based on it is reasoned either. The relation between weighted RB curve and rational Bézier curve is also discussed. It is not difficult to find that weighted RB curve can partly supply the gap of traditional rational Bézier curve in weight factor selecting. So the flexibility of curve design is improved. In the end, some available conclusions and visual consequences are obtained.