Blending algebraic surfaces

Reviewer: Michael A. O'Connor

If two surfaces are implicitly defined by the polynomials A and B , and two curves g and h on the surfaces are implicitly defined by the auxiliary polynomials G and H , then a surface implicitly defined by a polynomial C smoothly joins A at curve g to B at curve h if C is zero on g and h , the tangent planes to A and C agree along g , and the tangent planes to B and C agree along h . Such a surface is called a blending surface. Since the gradient to a surface determines its tangent plane, it is easy to see that for any polynomials P , Q , R , and S , the truly horrific general polynomial PAB + QAH 2 + RBG 2 + SG 2 H 2 defines a blending surface. What is not easy to see is that all implicitly defined blending surfaces must be of this form. This paper shows that this must be the case under certain technical assumptions, and discusses the relationships of many blending surface problems to membership in polynomial ideals (sets of polynomials generated in a manner similar to that above). The paper is peppered with simple examples and some pretty pictures and is basically well written. Since the major result is the generation of full ideals of blending functions, it is regrettable that this knowledge is never applied; only simple subsets that could be (and have been) generated independently are used. As a practical matter, the technical assumptions needed to apply the results seem difficult to enforce.