Rational Parametrization Of A Kind Of Blending Algebraic Surfaces

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 basic 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 basic 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 basic 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 G~1 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+vh1h2,where u = b2hi+b1h2 and deg(v) < 1.Evidently,the line S(h1, h2), so called the basic line, is on the blending plan.Let a plan rotate from h1 to h2 circled by the basic line, we will get a pencil of planes with the same axis ,each of the plan intersect f by the basic 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 parametrizationfor 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 Y{ , which we call the Hilbert -|- Burch matrix, and a related 3x3 matrix of linear forms in variables Xi.And the steps passing from Implicit Surface to cubic parametrizations are the following:{Implicit Surface} —> {3x3Matrix} —> {Hilbert—Burch Matrix} —t { Parametrizati on }First we turn the implicit equation .F = 0toa3x3 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 — x — t\y and n = x — tiyxn.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(ti) = m^rr?^, Q(i2) = riin2. By Theorem 2.3 we get the form of U:771 0 77iiU = | 0 n mpfor p = PiX + p2y + p$z + p4 and ki, k2 are constants. If t = a and t = a are complex conjugate roots of D(t) — 0, then tritangent plane m — x — ay, m = x — ay. The residual quadratic Q(a) in m factors into mim2 and <5(a) in m factors into Tn1m2-We form/ ? 0 777,i ^\ i\>iU,& n, 111/2 * jTo construct an equivalent real 3x3 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(—U) = F.The product U(s, t, w)T is a 3-tuple whose entries are linear in (s, v), t) and also linear in (x,y,z). Thus U(s,t,w)T can be rewritten H(x,y,z,l)T .where H is a 3 x 3 matrix whose entries are linear forms in (x,y,z).which multiply the 3x3 submatrices of H is the parametrization of S(F).by the form:While the basic line scan parametrization method can't find the uniform and rational parametrization .Berry's parametrization method is hard to find the first line and hard to get a precise parametrization. Berry's parametrization base on the basic line inherit their strongpoints, conquer their disadvantage.We are interested is the parameterization for the cubic blending surfaces . We find that the basic line S(hi:h2) is just one line on the surface by which we successfully complete the most difficult step of parameterization for...