中文摘要：

1引言基于多项式空间span{1,t,t^2,…,t^k}的B样条和Bézier曲线（面）是构造自由曲线、曲面强有力的工具,但是它们不能精确表示圆弧、椭圆等,也不能精确地表示正弦曲线和二次曲面,于是文献[1 提出了一种新的三次曲线（面）模型,称为C曲线（面）,它们是低次多项式样条曲线（面）的拓广,具有很多B样条的良好性质,如对称性、保凸性等,不仅能精确表示二次曲线和曲面和某些超越曲线,而且克服了NURBS求导求积复杂的困难,因此引起了国内外广泛的关注,近年来涌现了大量的文献.

英文摘要：

C-B spline curves and surfaces based on the space spanned by {sint, cost,, 1, t,…, t^k-3, } are new tools for surface modeling . In this paper a new explicit form for C-B spline curves and surfaces is proposed, which is represented as a linear combination of {sint, cos t,, 1, t,…, t^k-3} . And the explicit expression for its important subclass C-Bezier curves and surfaces is derived as well. Then the explicit form is used for the approximate degree reduction of C- Bezier curves and surfaces. The degenerate conditions for them are obtained. And from the new form a set of generated Hermite basis of order n can be expressed as a linear combination of C-Bezier basis of the same order. It is easy to see from the applications above that the use of this new different representation for C-B spline curves and surfaces in this paper is more convenient and effective in some cases. Finally, we give an example of C-Bezier curves of order 5 in the explicit representation.