Construction Of Blended Algebraic Surface With High Geometric Continuity

One of the central questions in CAGD is blending of surfaces. Recently,many results in commutative algebra and algebraic geometry can be calculatedbecause of the work of Groebner and Wu, which in turn helped the research ofimplicit algebraic surfaces. In this article, we will study the blending of implicitalgebraic surfaces with the intersections on planes by using algebraic geometry.First, we consider G² blending of two quadratic surfaces.Hypothesis: Let g₁ and g₂ be irreducible quadratic polynomials, whichdetermine two distinct surfaces, and let h₁ and h₂ be linear function, whichdetermine two different planes. Assume that S(g_β) and S(h_β) intersecttransversely with an irreducible quadratic intersection (β=1,2).Under this hypothesis, we let polynomial f of degree 4 such thatf=u₁g₁+a₁g₂+a₂h₂³ (1)Andu_β = u_β1h₁²+u_β2h₂²+u_β0h₁h₂ β =1,2 ( 2 )Where u_β1, u_β2,u_β0≠0 and a₁ ,a₂ are linear functions. We have thefollowing:Theorem 1 Let S ( h₁ )≠ S(h₂), if (1)is tenable,and u1 ,u2are held by(2),then there exists λ ≠0 and linear functions b₁ ,b₂, c₁ ,c₂,such thatg₁ + b₁h₁=λ g₂+b₂h₂, (3)(u₁₁h₁+ u₁₀h₂)g₁+c₁h₁²=(u₂₁h₁+u₂₀h₂)g₂+c₂h₂² ( 4 )Theorem 2 Let S ( h₁ )≠ S(h₂). Suppose that there exists quadraticpolynomial g, such thatg = g₁ +b₁h₁=λ g₂+b₂h₂ (5)If there exists real numbers μ , ω∈ R ,μω≠0, such that b1 ,b2 satisfy thefollowing constraint:h 2 + μh 1=ω(b 1+μb2), ( 6 )Then there exists polynomial f of degree 4 as following, such that the surface S(f)and S(gβ) meet with G2 continuity along the curve S(gβ, hβ)f = ug?vh1h2 (7)whereu = ω ( μ2 h12+h22?μh1h2) ( 8 )v = μ 2 h12?ω( μh1?h2)b1=h22?μω(h2?μh1)b2 ( 9 )Theorem 3 Suppose S ( h1 )∩ S(h2)≠φ, there exists polynomial f of degree4 as (1), and u1 ,u2are hold by(2), such that the surface S(f) and S(gβ) meet withG2 continuity along the curve S(gβ, hβ) if and only if there exists real numbersλ ≠ 0, μ≠0 and??????+===000211202100100λλλλλλμμμgggggg( 1 0 )Where g iλj are the expansion coefficients of g 1 ? λg2 in terms of h1 ,h2.Moreover, S(f) is defined by (7),(8),(9).Theorem 4 Suppose S ( h1 )∩ S(h2)=φ and denote h 2 = h1+δ, there existspolynomial f of degree 4 as (1), and u1 ,u2are hold by(2), such that the surfaceS(f) and S(gβ) meet with G2 continuity along the curve S(gβ, hβ) if and only ifthere exists real numbers λ ≠ 0, μ≠0 and??????+++=+=+=(1)(1)0(1)()02212010λλλλλλδδμμμδγggggggt(11)Where t ∈ Span( N(