| The process of converting the algebraic surface into the parametric surface is called parametrization , which has received much attention in recent years.In this paper,we look back two kinds of parametrization method: the base line scan parametrization and Berry's parametrization.Then a new method is given after we unite the strongpoints of the two kinds of method,we call it Berry's parametrization base on the base line. And this method can parametrize a kind of cubic algebraic surface, which plays an important role in the field of implicit algebraic surface modeling in CAGD. The base line scan parametrization method was introduced by Wang xianghai,Li chenlai.They use the method to parametrize the 2-way blending plans,before discuss ,we will explain the 2-way blending plan first. Let g1,g2 be quadratic polynomials which define quadrics S(gi)(i = 1, 2), hi∈R[x, y, z](i = 1,2) be two linear funcitons which determine different planes S(hi)(i = 1,2). Assume S(gi) and S(hi),intersect transversally along an irreducible planar quadratic curve S(gi,hi) (i = 1,2). One needs to construct a polynoimal f of degree 3,such that the surface S(f) and S(gi) meet with G1 continuity along the curve S(gi, hi)(i = 1,2)..By the study of CAGD group of Jinli University,the blending plan f can defined by f = ug+vh12,where u = b2h1+b1h2 and deg(v)≤1.Evidently,the line S(h1,h2), so called the base line, is on the blending plan.Let a plan rotate from h1 to h2 circled by the base line, we will get a pencil of planes with the same axis ,each of the plan intersect f by the base line and a quadratic curve.After parametrize all the quadratic curves ,we finally gained the parametrization of f.As we see,we can't find the uniform and rational parametrization by this method. Berry's parametrization method unify implicitization and parametrization for cubic surfaces by describing a sequence of steps that can be used to pass back and forth between the two descriptions. The parametrizations are in terms of polynomials of degree three. The key point is the construction of two matrices, a 3 x 4 matrix of linear forms in variables Yi , which we call the Hilbert Burch matrix, and a related 3×3 matrix of linear forms in variables Xi.And the steps passing from Implicit Surface to cubic parametrizations are the following: {Implicit Surface}→{3×3Matrix]→{Hilbert—Burch Matrix]→{Parametrization} First we turn the implicit equation F = 0 to a 3×3 matrix U. We do this by finding some lines on S.But this is not necessarily easy for the first line. Henceforth we assume we have found a line on S. We use this line to find some other lines on S. First change coordinates so that the line is x = y = 0. Rotate a plane about this line and intersect it with the surface. This is done by letting x = ty in F = 0 and cancelling the factor y that always appears. The other factor, Q(t) = 0, is quadratic in y, z and represents the residual intersection of the plane with the surface. To find some other lines on the surface, we look for values of t for which Q(t) factors into two lines in the plane x = ty. For this we take the discriminant: let the determinant of the Hessian of Q(t) is D(t). It will have degree 5 in t . Each of the 5 roots corresponds to a plane in which the residual intersection is degenerate. Some of the 5 roots are real, some are complex, so some of the 5 planes are real, some are complex. In the real planes some of the factorizations of Q(t) are real, some are complex.IF we choose two real roots t1, t2 of the discriminant of Q(t). Let m = xï¼t1y and n = xï¼t2y,m, n are real tritangent planes of S(F) (Any plane that contains three lines that lie in a surface is a tritangent plane.),and in each of these we have a real factorization into linear factors: Q(t1) = m1m2,Q(t2) = n1n2. By Theorem 2.3 we get the form of U: for p = p1X + p2y + p3z + p4 and k1, k2 are constants. If t =αand t = (α|—) are complex conjugate roots of D(t) = 0, then tritangent plane m = x ?αy,(m|—) = x ? (α|—)y. The residual quadratic Q(α) in m factors into m1m2 and Q((α|—)) in (m|—) factors into (m1m2|—).We formTo construct an equivalent real 3? matrix, we form U from (U|? by adding row 2 to row 1 and column 2 to column 1, subtracting one half row 1 from row 2 and one half column 1 from column 2, and multiplying row 2 and column 2 by i. This matrix U is real and det(æ¡¿) = F. The product U(s,t,w)T is a 3-tuple whose entries are linear in (s,w,t) and also linear in (x,y,z). Thus U(s,t,w)T can be rewritten H(x,y,z,1)T ,where if is a 3? matrix whose entries are linear forms in (x,y,z).which multiply the 3? submatrices of H is the parametrization of S(F).by the form: x = |H1|;y= |-H2|;z = |H3|;v = |-H4|;Based on the Berry's parametrization method and the base line scan parametrization method Cheng honglu give a new method which was called Berry's parametrization method based on the base line.with the new method, we could get the uniform and rational parametrization and also can conquer the difficult which when we use the Berry's parametrization method,we will be met.but the method has the disadvantage that it can be only used to the blending cubic surface whose equation do not have the term of yz, z. In this paper, based on the Berry's parametrization method based on the base line.we succeed to prove that the method also can be used to any of the blending cubic surface whose f = u1g1 + a1h12 = u2g2 + a2h22(ui∈I =1,h2>).Without losing the generality, let Q{h1,h2,t) = a11h22 + 2a12h2h3 + 2a13h2 + a22h32 + 2a23h3 + a33 and we could easily find that I2 (t) can be express a product of a simple polynomial and a quartic polynomial.so we could easily to get the root of the I2(t).Also as we analysis some blending cubic surfaces whose equation do not have the term of yz, z.we can find that with some special lack-term blending cubic surfaces,the root t of I2(t) are the same value.Proposition if blending cubic surfaces whose S(f) have the following property :1.Q(y, z; t) do not has the term of yz, y2.I2 = I3 = 03.g1,g2 are the same condition we can get the root t1 = i, t2 = -i of the I2(t) Then we will get the fixed factorization of Q(t) and the fixed Matrix U.A exactly rational parameterization of the cubic will be found at last. |
