The Graphic Solution Of Higher Smoothly Blending Of Quadric And Cubicimplicit Algebriaic Surface Along Planar Sections

Blending of implicit algebraic surfaces is one of the elementary question in CAGD. It is an important role in the theory research and application. 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 WenJun studied the problem by using the characteristic set method and changed them into the calculation of reducible rise set of multinomial equation combination. But the algorithm is complicated if used for the application of blending of general surfaces. In 1996, on the basis of Tieru Wu's study,Xuchao Wang gave method of GC¹ blending of two surfaces. 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. In 2007, in Yanfei Ren's master degree thesis, using above method, piloted study to higher smoothly blinding of three implicit quadratic algebraic surfaces.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, ifS(f) and S(g) being tangency at S(g,h), that f∈3 >, so for quadric and cubic implicit algebraic surfaces, we can conclude that blending surface G∈i,H_i^,3 >,G∈1,H₁³ >∩2,H₂³ >∩3,H₃³ >,or G = S₁G₁ +T₁H₁³ = S₂G₂ +T₂H₂³ = S₃G₃+T₃H₃³, degrees of S_i,T_i are determined byG_i,H_i (i= 1,2,3), If conform the form of S_i,T_i, then conform the form of blending surfaces. And then it can be converted into homogeneous linear equation combination.Using Maple, we can give the condition of the GC² blending of quadric and cubic implicit algebraic surfaces, the essential results are as follows:Theorem 3.1: To a quadratic algebraic surfaces and a cubic algebraic surfaces, cubic blending surfaces exist if and only if the coefficient matrix M₁ of somelinear equations satisfy Rank (M₁) < 7 .Theorem 3.2: To a quadratic algebraic surfaces and a cubic algebraic surfaces, four degrees blending surfaces exist if and only if the coefficient matrix M₂ of some linear equations satisfy Rank (M₂) < 22.Theorem 3.3: To a quadratic algebraic surfaces and a cubic algebraic surfaces, five degrees blending surfaces exist if and only if the coefficient matrix A of some linear equations satisfy Rank (A) < 50.Theorem 3.4: To Two quadratic algebraic surfaces and a cubic algebraic surfaces, cubic blending surfaces exist if and only if the coefficient matrix M₃ of somelinear equations satisfy Rank (M₃) < 12.Theorem 3.5: To Two quadratic algebraic surfaces and a cubic algebraic surfaces, four degrees blending surfaces exist if and only if the coefficient matrix M₄ ofsome linear equations satisfy Rank (M₄) < 36.Theorem 3.6. To Two quadratic algebraic surfaces and a cubic algebraic surfaces , five degrees blending surfaces exist if and only if the coefficient matrix A of some linear equations satisfy Rank (A) < 80.Theorem 3.7: To Two cubic algebraic surfaces and a quadratic algebraic surfaces, cubic blending surfaces exist if and only if the coefficient matrix M₅ of somelinear equations satisfy Rank (M₅) < 9.Theorem 3.8: To Two cubic algebraic surfaces and a quadratic algebraic surfaces, four degrees blending surfaces exist if and only if the coefficient matrix M₆ ofsome linear equations satisfy Rank (M₆) < 30.Theorem 3.9: To Two cubic algebraic surfaces and a quadratic algebraic surfaces, five degrees blending surfaces exist if and only if the coefficient matrix A of some linear equations satisfy Rank (A) < 70.