CHAPTER V - Spline Surfaces

Abstract

This chapter focuses on spline surfaces. Two-dimensional extensions of Bézier curves and B-spline curves are Bézier surfaces and B-spline surfaces, which are tools mainly for approximation. The most significant two-dimensional partition is the triangulation. There are many ways to construct bivariate spline functions with continuous first partial derivatives with respect to the triangulation. However, the degree of the spline function will rapidly increase if continuity of partial derivatives higher than first order is required. Just like univariate cubic spline functions, when bicubic spline functions are used for geometric design, the small deflection of data should be assumed; otherwise the fairness of the resulting surface may not be guaranteed. Modifications to bicubic functions are needed to fit large deflection or multivalued surfaces. The most successful modification is that the bicubic functions are replaced by vector-valued functions that are bicubic with respect to two parameters. Across the common boundary curve of two patches, sometimes the positions, slopes and curvatures, and even higher partial derivative vectors, should be matched. These problems have been solved by general Coons patch schemes.