Chapter 19 - Blending several implicit algebraic surfaces

Abstract

This chapter summarizes some basic results achieved on blending implicit algebraic surfaces. Due to the development of the theory of multi-spline functions, the theory of parametric and explicit surfaces has been studied extensively. It is found that the problem of determining the blending surface can be transformed into the computation of syzygy. A general blending surface with so-called ruled surfaces give the concrete expressions of the blending surface for blending quadratic surfaces, and the conditions for the existence of surfaces with lowest degree and its expressions are obtained. Under the basic hypothesis and the assumption of the existence of a quadric ruled surface, there exists an irreducible quartic polynomial. It is found that Warren's method can only produce quartic surfaces. This chapter also explains the construction of blending surfaces of low degree in general cases. The problem of blending two surfaces in two cases, and the necessary and sufficient conditions of existence of cubic blending surfaces and their expressions, respectively, are discussed. The relationship between the coefficients of the expansions and the conditions for the existence of the blending surfaces is also analyzed.