Higher Smoothly Blending And Graphic Solution Of Three Conicoid

Blending of implicit algebraic surfaces is one of the elementary problem in CAGD. It has important role in the theory research and application. But the problem didn't get the perfect results until the late 1970's, the constructive algebraic geometry made progress. In 1989, J. Warren described this kind of problem with ideal theory. In 1993, Wu Wen-Tsun studied the problem by using the characteristic set method and changed them into the calculation of irreducible ascending set of multinomial equations. But the algorithm is complicated if used for the application of blending of general surfaces. In 1996, on the basis of Tie-ru Wu's study, Xu-chao Wang gave a method of GC¹ blending of two conicoid . In 1998, Ju-xian Wang and Kai Yu used another method, gave the existence condition and realization method of GC⁰ blending and GC¹ blending of two conicoid . In 2001, Kai Yu converted problem of surfaces blending into solution of correlative homogeneous linear equations and avoided existing of quadratic control surface. In 2006, in Yun-dong Li's master degree thesis, using above method, piloted study to smoothly blending different quadric surfaces, and obtained the condition and algorithm of cubic and quadratic blending surface. But previously studies were almost low smoothly blending. By the method of computer algebraic, the higher smoothly blending of three quadratic surfaces is discussed. If g and h were two different irreducible polynomials, S(g) and S(h) across each at S(g,h), for every polynomial f , if S(f) and S(g) being tangency at S(g,h), that f∈< g,h³ >, so for three conicoid, we can conclude that blending surface G∈i,H_i³ >, G∈1,H₁³ >∩2,H₂³>∩3,H₃³>, orG = S₁G₁+T₁H₁³ =S₂G₂ + T₂H₂³=S₃G₃ + T₃H₃³, degrees of S_i,T_i aredetermined by G_i,H_i (i =1,2,3). The form of blending surface is determined by the form of S_i,T_i And then it can be converted into solution of correlative homogeneous linear equations. With Maple, we can give the condition of the GC² blending of three quadratic algebraic surfaces, the essential results are as follows:Theorem 1: Blending surface of three degrees is exist if and only if thematrix of correlative homogeneous linear equations satisfyRank(A)< 15 .Theorem 2: Blending surface of four degrees is exist if and only if thematrix of correlative homogeneous linear equations satisfyRank(A)< 24 .The last, we solve two examples of blending using computer algebraic system Maple and give their images, the fact show that the theory we give is effective to solve higher smoothly blending of three quadratic algebraic surfaces.