Blending And Rational Parametrization Of Algebraic Surface Families With Parameters

Blending and rational parametrization of surfaces are two fundamental problems in Computer Aided Geometric Design. Surface blending is to con-struct a transitional surface that smoothly joins two or more given solid models. The process of converting implicit algebraic representations of surfaces into ra-tional parametric representations of surfaces is called rational parametrization. This paper mainly studies blending and rational parametrization of algebraic surface families with parameters.An algebraic surface family with parameters means a family of surfaces denned by the zero set of a polynomial with parameters. We denote the real number field by R. Let X:={x,y,z} be the set of three variables,(?):={∈1,...,∈m} be the set of finite parameters. Denote R[(?)][X]:=(R[(?)])[X] as a polynomial ring over a polynomial ring IR[(?)].(?)e∈Rm, define the canonical specialization homomorphism σ∈:R[(?)][X]→R[X] induced by∈. Denoting the set of closed intervals over R. by R, the Cartesian product of m sets JR is denoted by IRm.Definition1(?)f∈R[(?)][X]; if let every e? vary in a closed interval, or written in the vector form as(?)∈(?):=I1×I2×...×Im∈IRm; then we obtain a family of polynomials {σ∈(f)∈R[X]:∈∈(?)}. We set V(f(?)):={V(σ∈(f)) (?), where V((?)) denotes the zero set of the polynomial σε(f) in R[X] and call V(f(?)) the algebraic surface family with parameters defined by f.Many problems can be described by algebraic surface families with param-eters. For example:on the one hand, in numerous engineering applications, people can only obtain surface data with measurement errors. However, for many singular problems, even tiny perturbation may cause the radical change of solutions. We can use algebraic surface families with parameters to rep-resent surfaces with measurement errors, and then discuss stable/continuous solutions of the new parametric problem with respect to these parameters; on the other hand, in geometric modelling, many human manufactured and nat-urally occurring objects have shell-like structures, that is, the object bodies consist of surfaces with thickness. Algebraic surface families with parameters also can represent surfaces with thickness, and their thickness can be controlled by adjusting the intervals of the parameters in their defining polynomials.The main results of this paper are as follows:(1) Blending of algebraic surface families with parameters.Lin and Rokne in2010considered the problem of blending algebraic in-terval surfaces. However, it is hard to precisely formulate the definition of ge-ometric continuity for algebraic interval surfaces. Due to interval arithmetic, the computed blending surfaces are thicker than the input surfaces. To over-come these drawbacks, the model of algebraic surface families with parameters is introduced in this paper.First, based on the definition of geometric continuity for algebraic surfaces, we propose the definition of geometric continuity for algebraic surface families with parameters.Definition2Let-V(f(?)) and V(g(?)) be algebraic surface families with parameters which intersect at an irreducible algebraic curve family with pa-rameters C, where f,g∈R[(?)][X]. We say that V(f(?)) and V(g(?)) meet with Gk rescaling continuity along C if 1.(?)∈∈(?), there exists a v∈(?), snch that V(σ∈(f)) and V(σv(g)) meet with Gk rescaling continuity along their common curve;2.(?)v∈(?), there exists a∈∈(?), such that V(σv(g)) and V(σ∈(f)) meet with Gk rescaling continuity along their common curve.The above definition guarantees that the computed blending surface fam-ilies match with the input surface families in thickness. Then we generalize the ideal theory method to algebraic surface families with parameters.Theorem1Let V(f(?)) and V(g(?),) be algebraic surface families with pa-rameters which intersect at an irreducible algebraic curve family with parame-ters C, where f,g∈R[(?)][X].(?)p1,p2∈R[(?)][X], we let f=p1g+p2hk+1∈<g,hk+1>(?)R[(?)][X], then the algebraic surface family with parameters V(f(?)) meets V(g(?)) with Gk rescaling continuity along C.When dealing with the blending problem, people usually choose the blend-ing surfaces of low degrees. However, the previous theorem doesn't specify how to compute such blending surface families. For this purpose, we formulate an algorithm IMPI for computing intersections of parametric polynomial ideals us-ing the comprehensive Grobner system method. The following subalgorithm SS is the Suzuki-Sato algorithm for computing comprehensive Grobner systems.Algorithm1IMPI(J1,...,Jr)Input: r ideals Ji=(Fi) in R[(?)][X], where Fi is the set of finite polyno-mials in R[(?)][X], i=1,...,r.Output: a comprehensive Grobner system g of the ideal (?) Ji in R[(?)][X].IMPI1: Let P←{1-(?)ti,t1F1,...,trFr}, and consider P as a subset of R[Ξ][t1,...tr,X].IMPI2: Let g'←ss(P,(?),(?)), where (?) is a block order with {t1,...,tr}>> X. IMPI3: Let g←(?). For every (Si,Ti,Gi')∈g', let Gi←{g:g∈Gt'∩R[(?)][X]}, g←g∪(si,Ti,Gi).IMPI4: Return g.Finally, we apply the algorithm IMPI to blend algebraic surface families with parameters. Suppose that g1,...,gr are distinct polynomials in R[(?)][X], h1,…,hr are polynomials in R[X], such that V(gi,(?)t) and V(ht) intersect at an algebraic curve family with parameters Ci, i=1,...,r.According to Theorem1, if f∈R[(?)][X] satifies f∈<g1, h1k+1>∩...∩<gr,hrk+1>(?) R[Ξ][X], then the algebraic surface family with parameters V(f(?)1∪...∪(?)r) meets V(gi,(?)i) with Gk rescaling continuity along Ci, i=1,...,r. Computing IMPl(<g1, h1k+1>,...,<gr, hrk+1?), we obtain a comprehensive Grobner system g={(S1, T1; G1),...,(Sl, Ti, Gl)} of the ideal <g1, h1k+1>∩…∩<gr,hrk+1>. Now, the parameter space Rm is divided into l segments V(Si)\V(Ti),i=1,...,l. Ve G V(Si)\V(Ti) and f∈<gi, h1k+1>∩...∩<gr, hrk+1>,σ∈(f) can be represented as a polynomial com-bination of σ∈(Gi). Thus we must make an additional judgement for g, that is, if there exists some V(Si0)\V(Ti0) such that (?)1∪…∪(?)r(?) V(Si0)\V(Ti0). If the above condition holds, then any algebraic surface familiy with parameters defined by the polynomial in <Gi0> is a solution; Otherwise, the algorithm fails.(2) Rational parametrization of quadric surface families with single pa-rameter.A quadric surface family with single parameter means that there is on-ly one parameter in its defining polynomial, and the parameter appears only once as well. Suppose that V(fI) is a quadric surface family with single pa-rameter defined by a quadric homogeneous polynomial f(x,y,z,w;∈), where X=(x, y, z, w)T is the homogeneous coordinate of the point on V(fI) and e is the parameter. We classify all quadric surface families with single parameter into two types: the ones that e appears in the w2term of f and the ones that e appears in the zw term of f. Obviously, for the quadric surface family with single parameter that e appears in the other term of f, we can transform it into one of the above types using a simple variable substitution.Based on the stereographic projection method, we propose the canonical form method CM to derive a rational parametrization for V(fI).Algorithm2CM(f(x,y,z,w;∈))Input: the defining polynomial f(x,y,z,w;∈) of V(fI).Output: a rational parametric representation of V(fI).CM1: Write f(x,y,z,w;∈) in the vector form as XTA∈X.CM2:(?)∈∈I, compute the corresponding protective transformation X P∈X':=P∈(x', y', z',w')T, where P∈is a nonsingular matrix with e, such that X'T(P∈TA∈P∈)X' is one of the following three canonical forms: C1x'2+y'2+z'2-w'2; C2x'2+y'2-z'2-w'2. C3x'2+y'2-z'2.CM3: Using the stereographic projection method, compute rational para-metric representations PC1,PC2,PC3(not uniquely) of C1,C2,C3.CM4:(?)∈∈I, a rational parametrization of V(fI) can be determined by substituting some PCi into the previous projective transformation X=P∈X'However, the parametrizations computed by the algorithm CM sometimes have discontinuities. The reason for this phenomenon is because when sim-plifying the coefficient matrix A∈with projective transformations, we treat∈and other coefficients in the defining polynomial equally. Actually, e which is the parameter must be distinguished from the others in the simplification process. Based on the above idea, we improve the canonical form method. The improved canonical form method may remove discontinuities. The following is the algorithm SCM for parametrizing V(fI) that e appears in the w2term of f.Algorithm3SCM(f(x,y,z,w;∈))Input: the defining polynomial f(x,y, z,w;∈) of V(fI).Output: a continuous rational parametric representation of V(fI) with respect to e.SCM1: Write f(x,y,z,w;∈) in the vector form as XTA∈X.SCM2: Compute the projective transformation X=PX':=P(x', y', z', w')T, where P∈is a nonsingular matrix without e, such that X'T(P∈TA∈P∈)X' is one of the following five canonical forms: SC1x'2+y'2+z'2+l(∈)w'2; SC2x'2+y'2-z'2+l(∈)w'2; SC3x'2+y'2+2a34z'w'+l(∈)w'2; SC4x'2-y'2+2a34z'w'+l(∈)w'2; SC5x'2+2a24y'w'+2a34z'w'+l(∈)w'2. where l(∈) is a linear polynomial of∈. Without loss of generality, we assume that a34is a nonzero real number.SCM3: Using the stereographic projection method, compute a rational parametric representation SPCi (not uniquely) of the corresponding canonical form SCi of V(fI).SCM4: a continuous rational parametric representation of V(fI) with respect to e can be determined by substituting the corresponding SPCi into the projective transformation X=PX'(3) Geometric information and rational parametrization of nonsingular cubic blending surfaces.Nonsingular cubic blending surfaces can also be viewed as special cubic surface families with parameters. Wang in2003discussed how to parametrize nonsingular cubic blending surfaces. However, the computed parametric repre-sentations are not rational. Wu and Cheng in2006discussed the parametriza- tion of the special nonsingular cubic blending surfaces denned by f=(b1(y-d2)+b2(x-d1))(x2+y2+z2-r2)-(x-d1){y-d2), where b1, b2, di, d2,r are parameters. The above results are further developed in this paper. We classify nonsingular cubic blending surfaces into two types: the specific forms and the general forms.For the specific forms, we analyse their geometric information, and prove thatTheorem2The specific forms of nonsingular cubic blending surfaces must be Fi(i=3,4,5) surfaces.To further determine surface types of the specific forms, we apply the complete discrimination system method proposed by Yang et al to the specific forms, and obtain the following theorem.Theorem3The discriminant sequence [Di,D2, D3,D4] of D(λ) of the specific forms is of the form [1,-8b4b2+3b32,16b42b2b0-18b42b12+14b4b3b2b1-6b4b32b0-4b4b23-3b33b1+b32b22,256b43b03-192b42b3b1b02+144b42b2b12b0-128b42b22b02-27b42b14+144b4b32b2b02-6b4b32b12b0-80b4b3b2b1b0+I8b4b3b2b13+16b4b24b0-4b4b23b12-27b34b02+18b33b2b1b0-4b33b13-4b32b23b0+b32b22b12], where bi is the coeffcient of λii in D(λ), i=0,1,...,4, respectively.,1. If one of the following conditions holds: D2<0∧D3<0∧D4>0; D2≥0∧D3≤0∧D4>0; D2<0∧D3≥0; D2=0∧D3>0, then the specific forms of nonsingular cubic blending surfaces are F3surfaces.2. If one of the following conditions holds: D2≤0∧Ds <0∧D4≤0; D2=0∧D3=0∧D4<0; D2>0∧D3<0∧D4=0; D2>0∧D4<0, then the specific forms of nonsingular cubic blending surfaces are F4surfaces.3. If one of the following conditions holds: D2>0∧D3>0∧D4≥0; D2≥0∧D3=0∧D4=0, then the specific forms of nonsingular cubic blending surfaces are F5surfaces.The above symbol "∧" indicates logical conjunction, which means that A ΔB holds if and only if both A and B hold simultaneously.Using the parametrization algorithm proposed by Berry and Patterson in2001, we obtain the uniform Hilbert-Burch matrix, the uniform related3×3real matrix, and the uniform rational parametric representation of the specific forms.Analogous conclusions also hold for the general forms.