The Constructive Theory Of A Kind Of Homogeneous G～2 Blending Algebraic Surfaces

One of the central question in CAGD is blending of surfaces, which provides the theoretical basis for the design technology of space surfaces. Because of the development of theory of multi-spline functions since 1970,s, the theory of parametric and explicit surfaces has been studied extensively. However, Its usefulness in the design of space surfaces is very limited. Implicit algebraic surfaces must be carefully studied, which requires the heavy machinery of algebraic geometry. But the constructive theory of classic algebraic geometry is so restricted that it can not be used efficiently to resolve many practical questions. Recently, many results in commutative algebra and algebraic geometry can be calculated because of the work of Groebner and Wu, which in turn helped the research of implicit algebraic surfaces. In this article we will study the constructive theory of a kind of homogeneous G~2 blending algebraic surfaces by using algebraic geometry. Basic Hypothesis .Let g 1and g 2be irreducible quadratic polynomials, which determine two distinct surfaces, and let h1 and h2 be linear functions, which determine two different planes. Assume that S ( gi) and S (h i)intersect transversely with an irreducible quadratic intersection (i =1,2). Firstly, we review 0G and 1G blending of quadratic surfaces, explore the theories of ruied surfaces and blending surfaces and two techniques are used in the process: the technique of ruied surfaces and the technique of standard expansion. Next, on the basis of the theories of 0G and 1G blendingsurfaces. We further consider the constructive of the blending of G 2algebraic surfaces with 2,3,4 and 5 degrees. We only consider when 1 1 1 1 3 2 2 2 23f = u g + a h = u g + a h (1) We obtain the conditions for the existence of 3,4degree mixture blending algebraic surfaces and construct the all possible surfaces,at the same time, we consider1 1 2 2k ki i iu = u h + u h( 0)iju ≠(2)k Theorem 1 There exists G 2 blending algebraic surfaces S ( f) if and only if there exist λ≠0, such that g1 =λg2 and it meets with S ( f)= S(g1 )=S(g2). Theorem 2 Let: 3 31 2 1 2 2 1 2 1u = αh , a = βg ;u = βh ,a = αg,we have f = αg 1h 23+βg2h13 Theorem 3 If s ( h1 )≠s(h2),then there exists homogeneous 4 degrees G 2 blending algebraic surfaces such as (1) and 2( 2) if and only if λ, μ≠0,such that ?????=+=++=+22112221221122111222bbhbhbbhbhgbhgbhμλ (3) Where b1 2,b21and b2 2are parametres. If (3) is tenable, without the consideration of nonzero constants ,We have ???????=?=?==+22121112122122211abbhabbhhhμλμμμ (4)It defines a group of homogeneous4 degrees G 2 blending algebraic surfaces in (1). theorem 4 If s ( h1 )≠s(h2),then there exists homogeneous 3 degree G 2 blending algebraic surfaces in (1) and 1(2) if and only if there existsλ, μ≠0,such that ?????=+=?++=+221122222112221111222bbhbhbbhbhgbhgbhμλ (5) in which b1 2,b21and b2 2are parametres. If (5) is tenable, without the consideration of nonzero constants ,We have ???????==?==+22223121112abaahhμμλμμμ (6) It defines a group of homogeneous 3 degrees G 2 blending algebraic surfaces Lastly, ,we provide specific methods of calculation and examples. Proposition 1 Under the basic hypothesis, if S (h 1)andS ( h2)are not parallel, there exists homogeneous 3 degrees G 2 blending algebraic surfaces homogeneous 3 degrees G 2 blending algebraic surfaces, such as (1) and 1(2) ,if and only if there exists λ≠0,such that 00 10 01211 20 020( ) 0s s ss s sλλλλλλ= = == ≠. Proposition 2 Under the basic hypothesis, if S (h 1)and S ( h2)are not parallel, there exists homogeneous4 degrees G 2 blending algebraic surfaces, such as (1) and 2(2) ,if and only if there exists λ≠0,such that 00 10 0120 02s s s0s sλλλλμλ= = == ? .