Bézier曲面的函数复合及其应用(英文)

Functional Compositions via Shifting Operators for Bézier Patches and Their Applications

目前有两种常用的 Bézier曲面片 ,分别称为三角和四边 Bézier曲面片 ,它们分别用不同的基函数表示 .本文通过移位算子和函数复合的方法 ,得到了两个关于这两种 Bézier曲面片的结果 .一个是四边 Bézier曲面片与一次三角 Bézier函数的复合 ,另一个是三角 Bézier曲面片与双线性四边 Bézier函数的复合 .在每一种情况中 ,复合所得到的 Bézier曲面片的控制顶点是原来 Bézier曲面片的控制顶点的线性组合 .移位算子的应用使得相应的推导过程变得简洁和直观 .这两个结果的应用包括 :两种 Bézier面片间的转化、裁剪 Bézier曲面片的精确表示。

There are two kinds of Bézier patches which are represented by different base functions, namely the triangular Bézier patch and the rectangular Bézier patch. In this paper, two results about these patches are obtained by employing functional compositions via shifting operators. One is the composition of a rectangular Bézier patch with a triangular Bézier function of degree 1, the other is the composition of a triangular Bézier patch with a rectangular Bézier function of degree 1×1. The control points of the resultant patch in either case are the linear convex combinations of the control points of the original patch. With the shifting operators, the respective procedure becomes concise and intuitive. The potential applications of the two results include conversions between two kinds of Bézier patches, exact representation of a trimmed surface, natural extension of original patches, etc.