Rational Parametric Formulas Of A Kind Of Cubic Implicit Algebraic Surface

Rational Parametric Formulas of a Kind of Cubic Implicit Algebraic SurfaceThe representation and modeling of surfaces are the basic problems in CAGD.To represent the surfaces,parametric form and implicit algebraic form are two main methods in the existing 3D CAD modeling system.They have their own inherent advantages and disadvantages, respectively.Although the geometric modeling based on the implicit algebraic surfaces has made considerable progress,but the representation of the implicit algebraic surfaces is still a key issues.The mathematic researchers of Jilin University have done a lot of research work in blending the implicit algebraic surfaces smoothly. They do the work under the following assumptions: Let gβ∈R[x,y,z}(β= 1,2) be two irreducible quadratic polynomials which define quadrics V(gβ)(β= 1,2); hβ∈R[x, y, z](β= l,2)be two irreducible linear polynomials which define planes V(hβ)(β= 1,2); V(gβ)与V(hβ)intersect transversally along an irreducible quadratic curve V(gβ, hβ)(β= 1,2).When clipping planes V(h₁) and V(h₂) are not parallel,we can expand g₁ and g₂ in terms of power products of h_i as the following formwhere detS_ij^β≤2 - (i+j) and S_ij^β∈Span_R(N(I)). Then the cubic surface V(f) meeting V(gi with GC¹ continuity along V(gi,hi) can be defined byAn special important situation is u_β=u_β1h₁ +u_β2h₂,β= 1, 2 (7)Berry's parametric method unify implicitization and parametrization for general cubic surfaces.The steps are the following:Implicitization (?) 3×3Matrix (?) HBMstrix (?) ParametrizationHere HB Matrix means the Hilbert-Burch Matrix. The main difficult is to get the 3×3Matrix U,whosc determinant is the implicitization of the surface.Find a line on the surface is the key issues of the first step.we assume we have found a line on the surface V(f).we use this line to find some other lines on V(f).we change coordinates so that the line is x = y = 0.Rotate a plane though the line intersect it with the surface,so we could get a conic.If the conic which is getted by intersect the plane with the surface degenerate into the product of the lines,we call it tritangent planes.This is done by letting x = ty in f = 0 and cancelling the factor y that always appears,so we can get a equation,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.I₃ will have degree 5 in t.We choose two real roots t₁,t₂ of I₃,and Q(t₁) = m₁m₂, Q(t₂) = n₁n₂.x = t₁y, x = t₂y are the tritangent planes. So we can get U by the following:Here P = p₁x + p₂y + p₃z + p₄, k_i, p_i are parametrics. After we get U,Though H(x, y, z, 1)^T = U(s, t, w)^T,we get 3 x 4 matrix H, the matrix we get after delete theβth column is noted by H_β, so we get the rational form of F:Based on the basic line,Cheng applied the Berry's parametric method to the parametric problem of the surface,and get the uniform parametric representation. But he have not solve the problem completely.In this paper,we first apply this method to the u_β∈I =< h₁,h₂ > surface, focus on the classification of the parametric process of the lack-term blending cubic surfaces,and list the parametric formulas for all the situations.In the process of getting tritangent planes,the conic Q(h₂,h₃;t) should degenerate into the product of the line when t get a certain value. HereThen wo have the following proposition:Proposition 1 If there exist a value t₀ oft that make the following equation valid:then the conic Q(h₂, h₃; t) can degenerate into the product of the line.Proposition 2 If there exist a value t₀ of t make the following valid,then the conic Q(h₂,h₃;t) can degenerate into the product of the line.But according to the above method,for the general blending cubic sur faces,there are two difficulties:1.the base line V(h₁,h₂) is not on the surface V(f).we get the line V(β₂h₂+l) which is to substitute the base line.2.The determinant of Q(h₁,h₂;t) Hessian matrix I₃ can not be degradated into quartic equation.And the quintic equation could not be solved in term of radicals.According to Galois Theory,we have the following:Corollary 1 A solvable,irreducible equation of prime degree in a field which is a subset of the real numbers has either one real root or all its roots are real.Now the problem converts to judge the number of real roots of the quintic equation.The Sturm Theorem of Algebra solves the problem. We offer some examples for either special situation or general situation.