Rational Parametrization Of Cubic Algebraic Surface Based On Sturm Group

Surface representation and modeling is one of the fundamental issuesin the area of computer-aided geometric design (CAGD),implicitization andparametrization of cubic surfaces have inherent advantages of their own. Al-though implicit algebraic surfaces based on geometric modeling has made con-siderable progress, but the explicit algebraic surfaces of the issue has alwaysbeen one basic theoretical issue to be further study of .Berry's parametric method unify implicitization and parametrization forgeneral cubic surfaces.The steps are the following:Implicitization?→3×3 Matrix ?→H B Matrix ?→ParametrizationAlthough very complete in theory,we always have di?cult to overcome thedi?culties encountered.The main di?cult is to get the 3×3Matrix U,whosedeterminant is the implicitization of the surface.Its element is linear combi-nation in (x,y,z).Find a line on the surface V(f).we use this line to find someother lines on V(f).We change coordinates so that the line is x = y = 0.Rotatea plane through the line intersect it with the surface degenerate into the prod-uct of the lines,we call it tritangent planes.This is done by letting x = ty inf = 0 and cancelling the factor y that often appears,so we get one equa-tion,note it by Q(y,z,t).When the determinant I3 of the Hessian Matrix ofQ(y,z,t)is zero,Q(y,z,t)can degenerate into the product of the lines.The de-terminant of Hessian matrix is one equation of four ranks in the symmet-ric form;while one equation of five ranks in the non-symmetric form.Solveingthe root of the equation is another di?cult problem in the first step of theprocess.We assume that the equation has two roots, they are a pair of con-jugate imaginary root t1,t1,They correspond to the decomposition of Q asQ(t1) = m1·m2,Q(t1) = m1·m2,wherem1,m2,m1,m2 is a polynomial of onerank.So x = t1y,x = t1y is a triple-section. Of course, we can get two real roots of I3.We also find the appropriate wayto make |U| = f.After getting the real matrix U,we compute H(x,y,z,1)T =U(s,t,w)T,and achieve 3×4 matrix H,let H removeing the firstβout to bethe matrix for Hβ,so the rational expression of f:In this paper, the form of symmetric surface in the absence of splicingof the circumstances at the time of the process parameters were discussed indetail,finding a straight line on the surface.The determinant of Hessian matrixis one four rank polynomial,The root of the equation and the multiplicity ofthe number of real roots of the following process parameters have a greatimpact.we know the decomposition of Q(h2,h3,t) is not always exist. Thereforeit is necessary for us to analyze the equation for the conditions of the existenceof virtual pairs of the root through the Sturm Group, So the first thing is todetermine whether there is re-root equation.we can see that the polynomial f(x) does not exist re-root if and only ifD4 = 0, of which D4 is the determinant for f of the discriminant matrix.Through the Sturm theorem, we get the conclusion:So that we can determine the existence of conjugate pairs of Hessianmatrix determinant to f of virtual root.For non-symmetrical form of splicing surface, there exist no general for-mula in genel case for the roots, to determine whether there is a re-root havegreat emphasis on the significance of the root.It is known from Proposition 0.3 that if f(x) does not exist multiple roots,the equation can be reduced in order to extract a root formula used to deriveall the solutions of equations.By Theorem 0.2, we can see that the polynomial f(x) does not exist re-root it is necessary and su?cient condition for D5 = 0, of which D5 is the determinant of the discriminant matrix for f.We have found from the following second ?at surface and cut-o? surfaceformed by splicingObtained by Hessian matrix determinant of the existence of multipleroots, so that we can derive all of its solutions. In this article, we give severalsurface parameters example in the forms of symmetric and non-symmetric intheir own.