Functional Compositions via Shifting Operators for Bézier Patches and Their Applications

There are two kinds of Bezier patches which are represented by different base functions, namely the triangular Bezier patch and the rectangular Bezier 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 Bezier patch with a triangular Bezier function of degree 1, the other is the composition of a triangular Bezier patch with a rectangular Bezier 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 Bezier patches, exact representation of a trimmed surface, natural extension of original patches, etc.