含参代数曲面族的光滑拼接及有理参数化
摘要
曲面的光滑拼接和有理参数化是计算机辅助几何设计中的两个基本问题.构造过渡曲面来光滑地连接两个或者多个实体模型这一过程称为拼接.由曲面的隐式代数表示转换成有理参数表示这一过程称为有理参数化.本文主要研究含参代数曲面族的光滑拼接和有理参数化.
所谓含参代数曲面族是指由含参数的多项式的零点集定义的代数曲面族.令R表示实数域,X：={x,y,z}是三个未定元构成的集合,(?)：={∈1,...,εm}是有限个参数构成的集合.记R[(?)][X]：=(R[(?)])[X]是多项式环R[(?)]上的多项式环.(?)E∈Rm,定义由∈导出的标准特定化同态σ∈：R[(?)][X]→R[X].R上所有闭区间构成的集合记为IR,m个IR的笛卡尔乘积记为IRm.
定义1(?)f∈R[(?)][X],如果每个∈i均在各自的定义区间内连续变动,即∈i∈Ii∈IR,i=1,...,m,或者将其写成向量形式,即(?)∈(?)：=I1×I2×…×Im∈IRm,那么我们可以得到一族多项式{σ∈(?)∈R[X]：E∈3}.令V(?)：={V(σ∈(f))：E∈3},这里V(σ∈(f))表示R[X]中的多项式σ∈(f)的零点集,我们称V(?)是由f定义的含参代数曲面族.
含参代数曲面族模型可以用来描述计算机辅助几何设计中的很多问题.例如：一方面,在众多工程应用中,人们获取的曲面数据通常是含误差的.然而,对于很多奇异性问题而言,微少的扰动亦能使问题的解发生本质性的变化.这时,我们可以使用含参代数曲面族表示含误差的曲面,那么我们只要讨论这个新的含参问题的关于其参数稳定/连续的解即可.另一方面,在几何造型中,人们经常会遇到类似于贝壳结构的物体,即物体是由带厚度的曲面片构成的.含参代数曲面族亦可以表示带厚度的曲面,其厚度可以通过调整各参数的定义区间来控制.
与代数区间曲面模型相比,含参代数曲面族模型的范围更广,并且不会出现由于区间运算所导致的“膨胀”；与等距代数曲面模型相比,含参代数曲面族模型的计算更加简单,但是如何自由地控制其厚度是一个比较困难的问题.
本文的主要工作可以分为下面三个部分.
(1)含参代数曲面族的光滑拼接.
2010年,林群和]R,okne讨论了代数区间曲面的拼接问题.然而,他们未能给出代数区间曲面的几何连续性的精确定义,并且由于区间运算会导致运算结果发生“膨胀”,他们构造的拼接曲面比输入曲面“厚”.本文的模型含参代数曲面族—可以有效地改进上述问题.
首先,我们将Warren在1989年提出的代数曲面的rescaling连续的定义推广到含参代数曲面族情形.
定义2假设含参代数曲面族V((?))和V((?))相交于一不可约的含参代数曲线族C,这里f,g∈R[(?)][X].称V((?))和V((?))沿着C Gk rescaling连续,如果
1.(?)E∈(?),存在v∈￡,使得V(σ∈(?))和V(σv(g))沿着其公共交线Gk rescal-ing连续；
2.(?)v∈￡,存在E∈(?),使得V(σv(g))和V(σ∈(f))沿着其公共交线Gk rescal-ing连续.
上述定义保证了我们构造的拼接曲面族和输入曲面族两者的厚度恰好“匹配”.接下来,我们将拼接理论中的理想论方法推广到含参代数曲面族情形.
定理1假设含参代数曲面族V(g(?))和代数曲面V(h)相交于一含参代数曲线族C,这里g∈R[(?)][X],h∈R[X].(?)p1,p2∈R[(?)][X],我们令f=p1g,p2hk+1∈(?)[(?)][X],那么含参代数曲面族V(f(?))和V(g(?))沿着C Gk rescaling连续.
人们在处理曲面拼接问题时,通常尽可能选择低次数的拼接曲面.然而,上述定理并未给出计算低次数拼接曲面族的方法.所以,下面我们使用全局Grobner系统方法,提出计算多个含参多项式理想的交的算法IMPI,这里算法SS是计算全局Grobner系统的Suzuki-Sato算法.
算法1IMPI(J1,...,Jr)
输入：R[(?)][X]中的r个理想Ji=,这里Fi是R[(?)][X]中的有限个多项式构成的集合,i=1,...,r.
输出：R[(?)][X]中理想(?)Ji的一个全局Grobner系统9.
IMPRI1:令P←{1-(?)ti,t1F1,...,trFr},再将P视为R[(?)][t1,...,tr,X]的子集.
IMPI2:令(?)←SS(P,(?)),这里(?)是满足{l1,...lr}》X的分块序.
IMPI3:令(?)←(?).对于每个(Si,Ti,Gi')∈g',令Gi←{g:g∈Gi'∩R[(?)][X]}, g←g∪(Si,Ti,Gi).
IMPI4:返回9.
最后,我们将算法IMPI应用到含参代数曲面族的拼接问题中.假设g1,...gr是R[(?)][X]中互异的多项式,h1,....,hr是R[X]中的多项式,且满足V(gi,(?)i)和V(hi)相交于一含参代数曲线族Ci,i=1...,r.
根据定理1,如果f∈R[(?)][X]满足f∈∩…∩(?)R[(?)]+[X],那么含参代数曲面族V(f(?))和V(gi(?))沿着Ci Gk rescaling连续,i=1,....,r.我们计算IMPI(,...,),得到(g1,h1k+1>∩…∩的一个全局Grobner系统g={(S1,T1,G1),...,(Sl,Tl,Gl)}这时,参数空间Rm被分割成l个部分V(Si)\V(Ti),i=1,...,l.(?)∈∈V(Si)\V(Ti)和.f∈∩…∩,σ∈(f)均可表示成σ∈(Gi)的多项式组合.所以,这时我们必须对(?)再做一次判断,即是否存在某个V(Si0)\V(Tiv),满足(?)1∪…∪(?)r(?)V(Si0)\V(Ti0).如果上述条件成立,那么由任一f∈(Gi。>定义的含参代数曲面族V(f(?)..(?)r)均为所求；否则,算法失败.
(2)单参二次曲面族的有理参数化.
称一个含参二次曲面族是单参二次曲面族,如果其定义多项式中仅含一个参数并且该参数仅出现在多项式的某一项中.假设V(fI)是由关于x,y,z,ω的二次齐次多项式f(x,y,z,w;∈)定义的单参二次曲面族,这里X：=(x,y,z,ω)T是曲面族上点的齐次坐标,∈是参数.我们将全体单参二次曲面族分成两类：∈出现在f的ω2项中的和∈出现在f的zω项中的.显然,对于∈出现在f的其他项中的单参二次曲面族,我们可以通过简单的变量替换将其转化成上面形式之一
基于经典的球极投影法,我们提出了参数化V(fI)的标准型方法算法CM.
算法2CM(f(x,y,z,w;∈))
输入：V(fI)的定义多项式f(x,y,z,w;∈).输出：V(fI)的一个有理参数表示.
CM1：将f(x,y,z,w;∈)写成形如XTA∈X的向量形式.
CM2：(?)∈I,计算相应的射影变换X=P∈X':=Pe(x',y',z',w')T,这里P∈是含∈的非奇异矩阵,使得X'T(P∈TA∈P∈)X是下面三种标准型之一c1x'2+y'2+z'2-w'2; c2x'2+y'2+z'2-w'2; c3x'2+y'2-z'2
CM3：使用球极投影法,分别计算标准型C1,C2,C3的有理参数表示PC1, PC2,PC3(不唯一)
CM4：(?)∈I,将其对应的标准型Ci的有理参数表示,PCi代入到相应的射影变换X=P∈X'中即可.
然而我们发现,使用算法CM计算出的有理参数表示有时会在某一点处出现“不连续”现象.这种“不连续”现象之所以出现,是由于我们在化简系数矩阵A。时,将∈与定义多项式中其他项的系数“一视同仁”.实际上,由于∈是参数,我们应当将∈与其他项的系数区别开来,对其单独处理.基于上述想法,我们改进了标准型方法,改进的标准型方法可以修复这种“不连续”现象.下面我们仅给出参数化∈出现在f的ω2项中的V(fI)的算法SCM.
算法3SCM(f(x,y,z,w;∈))
输入：V(fI)的定义多项式f(x,y,z,w;∈).
输出：V(fI)的一个关于∈连续的有理参数表示.
SCM1：将f(x,y,z,w;∈)写成形如XTA∈X的向量形式.
SCM2：计算射影变换X=PX':=P(x',y',z',w')T,这里P是不含∈的非奇异矩阵,使得X'T(PTA∈P)X'是下面五种标准型之一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.这里l(∈)是关于∈的线性多项式.不失一般性,我们假设a34是不等于零的实数.
SCM3：使用球极投影法,计算V(fI)对应的标准型SCi的有理参数表示SPCi(不唯一).
SCM4：将V(fI)对应的SPCi代入到射影变换X=PX'中即可.
(3)非奇异三次拼接曲面的几何信息和有理参数化.
非奇异三次拼接曲面亦可视为特殊的含参三次曲面族.2003年,王相海讨论了如何参数化非奇异三次拼接曲面,然而其计算的参数表示不是有理的.2006年,伍铁如和程宏路讨论了由f=(b1(y-d2)+b2(x-d1))(x2+y2+z2-r2)-(x-d1)(y-d2)定义的特殊非奇异三次拼接曲面的有理参数化,这里b1,b2,d1,d2,r是参数.上述工作在本文中得到了进一步地推广和深化.我们将非奇异三次拼接曲面分成两类：特殊形式曲面和一般形式曲面.
对于特殊形式曲面,我们分析了其几何信息,并且证明了
定理2特殊形式曲面只能是Fi(i=3,4,5)曲面.
为了进一步判断特殊形式曲面的曲面类型,我们使用杨路提出的多项式判别系统方法得到了下面定理.
定理3特殊形式曲面的D(λ)的判别式序列[D1,D2,D3,D4]为[1,-8b4b2+3632,16b42b2b0-18b24b12+14b4b3b3b1-6b4b32b0-4b4b23-3b33b1+b32b22,256b43b03-192b24b3b1b02+144b25b2b12b0128b24b22b02-27b42b14+144b4b32b2b02-6b4b32b12b0-80b4b3b22b1b0+18b4b3b2b13+16b4b24b0-4b4b23b12-27b34b02-18b33b2b1b0-4b33b13-4b32b23b0+b32b22b12],这里bi是D(λ)的λi项的系数,j=0,1,...,4.
1.如果下面条件之一成立：D2<0∧D3<0∧D4>0;D2≥0∧D3≤0∧D4>0;D2<0∧D3≥0;D2=0∧D3>0,那么特殊形式曲面是F3曲面.
2.如果下面条件之一成立：D2≤0∧D3<0∧D4≤0;D2=0∧D3=0∧D4<0;D2>0∧D3<0∧D4=0;D2>0∧D4<0,那么特殊形式曲面是F4曲面.
3.如果下面条件之一成立：D2>0∧D3>0∧D4≥0;D2≥0∧D3=0∧D4=0,那么特殊形式曲面是F5曲面.这里“八”表示逻辑交,即A八B成立当且仅当A和B同时成立.
根据Berry和Patterson在2001年提出的非奇异三次曲面的有理参数化算法,我们得到了特殊形式曲面的统—Hilbert-Burch矩阵、相关的3×3实矩阵和有理参数表示.
对于一般形式曲面,我们亦有类似的结论成立.
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∈(?)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∈∩...∩(?) 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(,...,∩…∩. 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∈∩...∩,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 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 that
Theorem2The 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.
所谓含参代数曲面族是指由含参数的多项式的零点集定义的代数曲面族.令R表示实数域,X：={x,y,z}是三个未定元构成的集合,(?)：={∈1,...,εm}是有限个参数构成的集合.记R[(?)][X]：=(R[(?)])[X]是多项式环R[(?)]上的多项式环.(?)E∈Rm,定义由∈导出的标准特定化同态σ∈：R[(?)][X]→R[X].R上所有闭区间构成的集合记为IR,m个IR的笛卡尔乘积记为IRm.
定义1(?)f∈R[(?)][X],如果每个∈i均在各自的定义区间内连续变动,即∈i∈Ii∈IR,i=1,...,m,或者将其写成向量形式,即(?)∈(?)：=I1×I2×…×Im∈IRm,那么我们可以得到一族多项式{σ∈(?)∈R[X]：E∈3}.令V(?)：={V(σ∈(f))：E∈3},这里V(σ∈(f))表示R[X]中的多项式σ∈(f)的零点集,我们称V(?)是由f定义的含参代数曲面族.
含参代数曲面族模型可以用来描述计算机辅助几何设计中的很多问题.例如：一方面,在众多工程应用中,人们获取的曲面数据通常是含误差的.然而,对于很多奇异性问题而言,微少的扰动亦能使问题的解发生本质性的变化.这时,我们可以使用含参代数曲面族表示含误差的曲面,那么我们只要讨论这个新的含参问题的关于其参数稳定/连续的解即可.另一方面,在几何造型中,人们经常会遇到类似于贝壳结构的物体,即物体是由带厚度的曲面片构成的.含参代数曲面族亦可以表示带厚度的曲面,其厚度可以通过调整各参数的定义区间来控制.
与代数区间曲面模型相比,含参代数曲面族模型的范围更广,并且不会出现由于区间运算所导致的“膨胀”；与等距代数曲面模型相比,含参代数曲面族模型的计算更加简单,但是如何自由地控制其厚度是一个比较困难的问题.
本文的主要工作可以分为下面三个部分.
(1)含参代数曲面族的光滑拼接.
2010年,林群和]R,okne讨论了代数区间曲面的拼接问题.然而,他们未能给出代数区间曲面的几何连续性的精确定义,并且由于区间运算会导致运算结果发生“膨胀”,他们构造的拼接曲面比输入曲面“厚”.本文的模型含参代数曲面族—可以有效地改进上述问题.
首先,我们将Warren在1989年提出的代数曲面的rescaling连续的定义推广到含参代数曲面族情形.
定义2假设含参代数曲面族V((?))和V((?))相交于一不可约的含参代数曲线族C,这里f,g∈R[(?)][X].称V((?))和V((?))沿着C Gk rescaling连续,如果
1.(?)E∈(?),存在v∈￡,使得V(σ∈(?))和V(σv(g))沿着其公共交线Gk rescal-ing连续；
2.(?)v∈￡,存在E∈(?),使得V(σv(g))和V(σ∈(f))沿着其公共交线Gk rescal-ing连续.
上述定义保证了我们构造的拼接曲面族和输入曲面族两者的厚度恰好“匹配”.接下来,我们将拼接理论中的理想论方法推广到含参代数曲面族情形.
定理1假设含参代数曲面族V(g(?))和代数曲面V(h)相交于一含参代数曲线族C,这里g∈R[(?)][X],h∈R[X].(?)p1,p2∈R[(?)][X],我们令f=p1g,p2hk+1∈
人们在处理曲面拼接问题时,通常尽可能选择低次数的拼接曲面.然而,上述定理并未给出计算低次数拼接曲面族的方法.所以,下面我们使用全局Grobner系统方法,提出计算多个含参多项式理想的交的算法IMPI,这里算法SS是计算全局Grobner系统的Suzuki-Sato算法.
算法1IMPI(J1,...,Jr)
输入：R[(?)][X]中的r个理想Ji=
输出：R[(?)][X]中理想(?)Ji的一个全局Grobner系统9.
IMPRI1:令P←{1-(?)ti,t1F1,...,trFr},再将P视为R[(?)][t1,...,tr,X]的子集.
IMPI2:令(?)←SS(P,(?)),这里(?)是满足{l1,...lr}》X的分块序.
IMPI3:令(?)←(?).对于每个(Si,Ti,Gi')∈g',令Gi←{g:g∈Gi'∩R[(?)][X]}, g←g∪(Si,Ti,Gi).
IMPI4:返回9.
最后,我们将算法IMPI应用到含参代数曲面族的拼接问题中.假设g1,...gr是R[(?)][X]中互异的多项式,h1,....,hr是R[X]中的多项式,且满足V(gi,(?)i)和V(hi)相交于一含参代数曲线族Ci,i=1...,r.
根据定理1,如果f∈R[(?)][X]满足f∈
(2)单参二次曲面族的有理参数化.
称一个含参二次曲面族是单参二次曲面族,如果其定义多项式中仅含一个参数并且该参数仅出现在多项式的某一项中.假设V(fI)是由关于x,y,z,ω的二次齐次多项式f(x,y,z,w;∈)定义的单参二次曲面族,这里X：=(x,y,z,ω)T是曲面族上点的齐次坐标,∈是参数.我们将全体单参二次曲面族分成两类：∈出现在f的ω2项中的和∈出现在f的zω项中的.显然,对于∈出现在f的其他项中的单参二次曲面族,我们可以通过简单的变量替换将其转化成上面形式之一
基于经典的球极投影法,我们提出了参数化V(fI)的标准型方法算法CM.
算法2CM(f(x,y,z,w;∈))
输入：V(fI)的定义多项式f(x,y,z,w;∈).输出：V(fI)的一个有理参数表示.
CM1：将f(x,y,z,w;∈)写成形如XTA∈X的向量形式.
CM2：(?)∈I,计算相应的射影变换X=P∈X':=Pe(x',y',z',w')T,这里P∈是含∈的非奇异矩阵,使得X'T(P∈TA∈P∈)X是下面三种标准型之一c1x'2+y'2+z'2-w'2; c2x'2+y'2+z'2-w'2; c3x'2+y'2-z'2
CM3：使用球极投影法,分别计算标准型C1,C2,C3的有理参数表示PC1, PC2,PC3(不唯一)
CM4：(?)∈I,将其对应的标准型Ci的有理参数表示,PCi代入到相应的射影变换X=P∈X'中即可.
然而我们发现,使用算法CM计算出的有理参数表示有时会在某一点处出现“不连续”现象.这种“不连续”现象之所以出现,是由于我们在化简系数矩阵A。时,将∈与定义多项式中其他项的系数“一视同仁”.实际上,由于∈是参数,我们应当将∈与其他项的系数区别开来,对其单独处理.基于上述想法,我们改进了标准型方法,改进的标准型方法可以修复这种“不连续”现象.下面我们仅给出参数化∈出现在f的ω2项中的V(fI)的算法SCM.
算法3SCM(f(x,y,z,w;∈))
输入：V(fI)的定义多项式f(x,y,z,w;∈).
输出：V(fI)的一个关于∈连续的有理参数表示.
SCM1：将f(x,y,z,w;∈)写成形如XTA∈X的向量形式.
SCM2：计算射影变换X=PX':=P(x',y',z',w')T,这里P是不含∈的非奇异矩阵,使得X'T(PTA∈P)X'是下面五种标准型之一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.这里l(∈)是关于∈的线性多项式.不失一般性,我们假设a34是不等于零的实数.
SCM3：使用球极投影法,计算V(fI)对应的标准型SCi的有理参数表示SPCi(不唯一).
SCM4：将V(fI)对应的SPCi代入到射影变换X=PX'中即可.
(3)非奇异三次拼接曲面的几何信息和有理参数化.
非奇异三次拼接曲面亦可视为特殊的含参三次曲面族.2003年,王相海讨论了如何参数化非奇异三次拼接曲面,然而其计算的参数表示不是有理的.2006年,伍铁如和程宏路讨论了由f=(b1(y-d2)+b2(x-d1))(x2+y2+z2-r2)-(x-d1)(y-d2)定义的特殊非奇异三次拼接曲面的有理参数化,这里b1,b2,d1,d2,r是参数.上述工作在本文中得到了进一步地推广和深化.我们将非奇异三次拼接曲面分成两类：特殊形式曲面和一般形式曲面.
对于特殊形式曲面,我们分析了其几何信息,并且证明了
定理2特殊形式曲面只能是Fi(i=3,4,5)曲面.
为了进一步判断特殊形式曲面的曲面类型,我们使用杨路提出的多项式判别系统方法得到了下面定理.
定理3特殊形式曲面的D(λ)的判别式序列[D1,D2,D3,D4]为[1,-8b4b2+3632,16b42b2b0-18b24b12+14b4b3b3b1-6b4b32b0-4b4b23-3b33b1+b32b22,256b43b03-192b24b3b1b02+144b25b2b12b0128b24b22b02-27b42b14+144b4b32b2b02-6b4b32b12b0-80b4b3b22b1b0+18b4b3b2b13+16b4b24b0-4b4b23b12-27b34b02-18b33b2b1b0-4b33b13-4b32b23b0+b32b22b12],这里bi是D(λ)的λi项的系数,j=0,1,...,4.
1.如果下面条件之一成立：D2<0∧D3<0∧D4>0;D2≥0∧D3≤0∧D4>0;D2<0∧D3≥0;D2=0∧D3>0,那么特殊形式曲面是F3曲面.
2.如果下面条件之一成立：D2≤0∧D3<0∧D4≤0;D2=0∧D3=0∧D4<0;D2>0∧D3<0∧D4=0;D2>0∧D4<0,那么特殊形式曲面是F4曲面.
3.如果下面条件之一成立：D2>0∧D3>0∧D4≥0;D2≥0∧D3=0∧D4=0,那么特殊形式曲面是F5曲面.这里“八”表示逻辑交,即A八B成立当且仅当A和B同时成立.
根据Berry和Patterson在2001年提出的非奇异三次曲面的有理参数化算法,我们得到了特殊形式曲面的统—Hilbert-Burch矩阵、相关的3×3实矩阵和有理参数表示.
对于一般形式曲面,我们亦有类似的结论成立.
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∈
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∈
(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 that
Theorem2The 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.
引文
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