Research And Realization Of Smooth Blending Technology Of Two Quadratic Surfaces

One of the elementary question in CAGD is blending of the implicit algebraic surfaces, which provides the theoretical bases for the design technology of space surfaces. It has important role in the theory research and application. Because of the breakthrough in the constructive algebraic geometry at the end of 1970's, it provides the possibility for the blending of the common algebraic surfaces. So many methods of constructive blending surfaces appear. But how to obtain the lowest degree blending surfaces is always the hot problem. Wu Wen-Tsun convert the problem of surfaces blending into the calculation of the irreducible ascending set of polynomial equations in 1993. This method can get the blending surface which degree probably is the lowest one. In the Wu'method, it doesn't make sure of the degree of blending surfaces in the first step of arithmetic, it adopt the trial_and_error method, namely, it assume that the blending surfaces is quadratic surface. If the equations have not nonzero solution in the fifth step of arithmetic, then increase the degree of the blending surface. As it is known , the coefficients of ternary quadratic algebraic equation are 10; when the degree of the algebraic surfaces rise to cubic, the coefficients are 20; if the degree is four, the coefficients are 35. Therefore, the cycle will be decrease and the space will be saved if the degree of the blending surface can be made sure before the performance of the arithmetic. In this paper, we discuss the problem of smooth blending of two quadratic surface in the two special conditions: one is the two cutting plane which upright with coordinate axis, the other is the two cutting plane parallel each other. wu'method can be mended in these two special instance in this paper. namely, we work out a arithmetic that it can automatic judge the degree of the blending surface. As a result we can make sure of the degree of the blending surface before the performance of the arithmetic in the [3]. So we can resolve equations directly whether the equations have nonzero or not. Arithmetic pre deg:make sure of the degree of blending surfaces g = 0。Input : α, a1 , b1 , d 0 , d1 , T1 , h1 , a2 , b2 , T2 ,h2 output:the degree of blending surfaces g = 0。step: V4. If ?1 is a constant, ? 2 is a constant,and ?1 +? 2 ≠0,then there is a cubic blending surface .else if ?1 is a constant, ? 2 is a constant,and ?1 +? 2= 0 ,then there is a quadratic blending surface. According to the arithmetic pre deg, we can make sure of the degree of blending surfaces beforehand in the two special instance. Therefore, we can make sure of the degree of blending surface at the first step in the wu'method.The following is the mending wu'method . Arithmetic blendsur :calculate the smooth blending surface of the two given quadratic surfaces. Input:the two given quadratic surfaces f1 = 0, f 2 = 0,and two cutting plane h1 = 0, h2 = 0 the algebra surface g = 0 that it's degree is k are undecided in coefficients。Output:the surface g = 0 that it's coefficients are ascertained. Step: 1.Make indeterminate x , y ,z suitability compositor vs1 ,and construct ascending set asi := triset ([ f i , hi 2], vs1) of polynomial set about indeterminate order, i = 1,2; 2. p1 := premas ( g , as1 , vs1), p2 := premas ( g , as 2, vs1); 3. c1 := coeffs ( p1 , vs1), c2 := coeffs ( p2 , vs1), cs := union ( c1 , c2) vs := the coefficients of the algebra surface g = 0; 4. roots ( cs , vs ); The blending surface is implicit algebraic equation that we obtain by arithmetic blendsur . The expression of the implicit surface is apt to distinguish whether a point is in the algebraic blending surface or not. It is easy to express a close body and the operation is close in the geometry operation. But it has many embranchments and its shape is difficult to control and arithmetic is not easy to implement. The surface in the parameter form is simple in its structure and the implementation is simple too. So it becomes a mainstream of geometry design. Therefore, for generating realistic images s of the given surfaces and blending surface, we should convert the implicit algebraic equation into parametric equation. Arithmetic parame :parametization of cubic implicit algebraic surface...