# The Viro Method for Construction ofCr Piecewise Algebraic Hypersurfaces

Authors: Yisheng Lai; Weiping Du; Renhong Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We propose a new method to construct a real piecewise algebraic hypersurface of a given degree with a prescribed smoothness and topology. The method is based on the smooth blending theory and the Viro method for construction of Bernstein-Bézier algebraic hypersurface piece on a simplex.


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## Body

## 1. Introduction

LetΔ be a simplicial subdivision of a region Ω in Rk. Δ is called a pure simplicial complex of dimension k and can be described as a finite collection of simplices such that the faces of each element of Δ are elements of Δ, and the intersection of any two elements of Δ is an element of Δ, and every maximal element of Δ (with respect to inclusion) is a k-dimensional simplex. We will sometimes refer to the m-dimensional elements of Δ as m-cells and the simplicial subdivision as a k-complex. If two k-dimensional simplices in Δ meet in a face of dimension k-1, we say they are adjacent. Δ is said to be hereditary if for every τ∈Δ (including the empty set) any two n-dimensional simplices S,S′ of Δ that contain τ can be connected by a sequence S=S1,S2,…,Sm=S′ in Δ such that each Si is k-dimensional, each Si contains τ, and Si and Si+1 are adjacent for each i (see [1, 2]).LetΔ be a pure, hereditary k-dimensional simplicial complex in Rk, let S1,S2,…,Sq be a given, fixed, ordering of the k-cells in Δ, and let Ω=⋃i=1q‍Si. Now, we recall the definitions of Cr(Δ) and Cnr(Δ) (see [1, 2]).Definition 1.
For a nonnegative integerr and a k-complex Δ, Cr(Δ) is the set of Cr functions f on Ω (i.e., functions such that all rth order partial derivatives exist and are continuous on Ω) such that, for every δ∈Δ including those of dimension <k, the restriction f|δ is a polynomial function f|δ∈R[x1,…,xk]. Cnr(Δ) is the subset of f∈Cr(Δ) such that the restriction of f to each cell in Δ is a polynomial function of degree k or less.It is clear thatCr(Δ) and Cnr(Δ) are a Noether ring and a finite dimensional linear vector space, respectively, and are called a Cr spline ring and a multivariate spline space with degree n and smoothness r, respectively. We call
(1)𝒵(f)∶={(x1,…,xk)∈Ω∣f(x1,…,xk)=0}
a real Crpiecewise algebraic hypersurface (see [1–3]), where f∈Cr(Δ).An important direction in real algebraic geometry during the last three decades is the construction of real algebraic hypersurfaces of a given degree with prescribed topology. Central to these developments is a combinatorial construction due to the Viro method [4–6]. The Viro method is a powerful construction method of real nonsingular algebraic hypersurfaces with prescribed topology (see [4–14]). It provides a link between the topology of real algebraic varieties and toric varieties. It is based on polyhedral subdivisions of the Newton polytopes. A particular and important case of the Viro method is called combinatorial patchworking; the combinatorial patchworking is a particular case of the Viro method which is characterized by the following two properties: the subdivision used is a triangulation, and each monomial of any “block” polynomial corresponds to a vertex of the Newton simplex.Roughly speaking, the Viro method starts with a convex (or coherent) polyhedral subdivision{Pi,i∈I} of a polytope P and a collection {fi,i∈I} of real nondegenerate polynomials fi with Newton polyhedra Pi whose truncations on common faces of Newton polyhedra coincide. Then, a Viro polynomial f with Newton polytope P is defined, and Viro’s theorem asserts that the topology of the real hypersurface Z(f) defined by f can be recovered by gluing together pieces of the real hypersurfaces Z(fi).It is well known that the hypersurface generally possesses complex topological or geometric structures in CAGD and geometric modelling. Moreover, the surface can be represented in Bernstein-Bézier form since it is often defined on a simplex, and writing a polynomial in it’s the Bernstein-Bézier representation has significant advantages since its coefficients reflect geometric information about the shape of the polynomial surface, and the barycentric coordinates relative to the simplex are affine invariant, and Bernstein-Bézier basis polynomials exhibit many important properties (see [3, 15–17]). Therefore, based on the Viro method and the Newton polyhedra of Bernstein-Bézier polynomial, Lai et al. in [18] established a new method for the construction of Bernstein-Bézier algebraic hypersurfaces on a simplex with a prescribed topology and presented a method to describe the topology of the Viro Bernstein-Bézier algebraic hypersurface piece.In CAGD and geometric modelling, most of the complex curves and surfaces are expressed by piecewise polynomials with certain smoothness (see [3, 17, 18]). Thus, the aim of this paper is to establish a new method for the construction of real piecewise algebraic hypersurfaces of a given degree with certain smoothness and prescribed topology. The new Viro method is based on the work of Lai et al. in [18] and the smooth blending theory.The paper is organized as follows. Section2 reviews briefly the Viro method for the construction of Bernstein-Bézier algebraic hypersurface piece on a simplex. In Section 3, we define the chart of the piecewise polynomial and deal with some properties of the chart of the Bernstein-Bézier polynomial and the chart of the piecewise polynomial. Section 4 is devoted to a new method for the construction of the real piecewise algebraic hypersurfaces of a given degree with certain smoothness and prescribed topology. The method is primarily based on the work of Lai et al. in [18] and the smooth blending theory.


## 2. Construction of the Bernstein-Bézier Algebraic Hypersurface Piece

This section reviews briefly the Viro method for the construction of Bernstein- Bézier algebraic hypersurface piece, as stated in [18].Throughout this paper, we denote byℝ+ (resp., ℝ+*) the set of real numbers x such that x≥0 (resp., x>0) and by ℤ+ the set of nonnegative integers. Let S=[v1,…,vk+1] will be a fixed k-dimensional simplex with vertices v1,…,vk+1.It is well known (cf. [3, 15]) that for any point p∈S, it can be expressed uniquely as
(2)p=∑j=1k+1‍τjvj,
where ∑j=1k+1τj=1,  τj≥0, j=1,2,…,k+1, and τ=(τ1,…,τk+1) is the barycentric coordinates of p with respect to S (we abusively confuse a point p=∑j=1k+1τjvj∈S with the corresponding barycentric coordinates τ=(τ1,…,τk+1)).Let(3)ℤ+(k,n)={λ=(λ1,…,λk+1)∈(ℤ+)k+1∣|λ|=∑i=1k+1‍λi=n}.We call a pointp∈S domain point if p=(λ1/n,…,λk+1/n) with (λ1,…,λk+1)∈ℤ+(k,n). Further on, polyhedron relative to the simplex S means a convex polyhedron in S with domain points as its vertices.It is well known (c.f. [3, 15]) that for any polynomial f over S with k variables and degree at most n, it can be represented in the Bernstein-Bézier form as follows:
(4)f(τ)=∑|λ|=n‍bλBλ,n(τ),
where bλ∈ℝ, λ=(λ1,…,λk+1)∈ℤ+(k,n),
(5)Bλ,n(τ)=n!λ1!⋯λk+1!τ1λ1⋯τk+1λk+1
are the Bernstein basis of degree n relative to S. The f(τ) is called a Bernstein-Bézier polynomial or B-form of the polynomial f relative to the simplex S. We will refer to the polynomials in B-form as BB-polynomials and to their coefficients bλ as BB-coefficients. The set of its zero points in S is called a Bernstein-Bézier algebraic hypersurface piece (BB-algebraic hypersurface piece for short).For the BB-polynomialf defined in (4), set
(6)μ(f)∶={(λ1n,…,λk+1n)∣λ∈ℤ+(k,n),bλ≠0}.
The convex hull conv(μ(f)) of μ(f) on S, denoted by N(f), is called Newton polyhedron (relative to S) of the BB-polynomial f.For a setΓ⊂S and a BB-polynomial f=∑|λ|=nbλBλ,n(τ), denote BB-polynomial ∑|λ|=n,λ∈ΓbλBλ,n(τ) by fΓ. It is called the Γ-truncation of f.A BB-polynomialf is called nondegenerate if f and any truncation fδ on a proper face δ of N(f) has a nonsingular zero set in (ℂ*)k+1.Let𝒥S be a subset of the set of domain points in simplex S and PS=conv(𝒥S). Define the moment map associated with 𝒥S relative to the simplex S, ϕ𝒥S:Int(S)→Int(PS), by
(7)ϕ𝒥S(τ)=∑(i1/n,…,ik+1/n)∈𝒥S‍τ1i1⋯τk+1ik+1(i1/n,…,ik+1/n)∑(i1/n,…,ik+1/n)∈𝒥S‍τ1i1⋯τk+1ik+1,
where Int(PS) is the complement in PS of the union of all its proper faces.Definition 2.
Letf be a BB-polynomial with a Newton polyhedron P. Then the closure of ϕμ(f)({τ∈Int(S),f(τ)=0}) in P is called chart CtS(f) of f, where ϕμ(f):Int(S)→Int(P) is the moment map associated with μ(f) relative to the simplex S.LetP,P1,…,Pm⊂S be polyhedra with P=⋃i=1m‍Pi and Int(Pi)⋂‍Int(Pj)=∅ for i≠j. Assume that ν:P→ℝ is a continuous, piecewise linear, nonnegative convex function satisfying the following conditions:(1)
all the restrictionsν|Pi are linear;(2)
if the restriction ofν to an open set is linear, then this set is contained in one of the Pi;(3)
ν(P⋂‍L(k,n))⊂ℤ, where L(k,n) is the set of the domain points.Then, this function ν with this property is said to convexify{P1,…,Pm}.Letf1,…,fm be BB-polynomials over ℝ in k+1 variables with N(fi)=Pi. Let fiPi⋂‍Pj=fjPi⋂‍Pj for any i,j. Then, there exists a unique BB-polynomial f with N(f)=P=⋃i=1m‍Pi and fPi=fi for i=1,…,m. If f(τ)=∑|λ|=nbλBλ,n(τ) and ν is a function convexifying {P1,…,Pm}, we put
(8)ft(τ)=∑|λ|=n‍bλBλ,n(τ)tν(λ/n).
The BB-polynomials ft are said to be obtained by patchworking BB-polynomials f1,…,fm by ν or, briefly, ft is a patchwork of BB-polynomials f1,…,fm by ν.Letf1,…,fm be BB-polynomials in k+1 variables with N(fi)=Pi, and let Int(Pi)⋂‍Int(Pj)=∅ for i≠j. A chart CtS(f) of a BB-polynomial f with N(f)=P is said to be obtained by patchworking charts of BB-polynomials f1,…,fm and it is a patchwork of charts of BB-polynomials f1,…,fm if P=⋃i=1m‍Pi and the chart CtS(f) of f, up to isotopy, is ⋃i=1m‍CtS(fi).The result about the Viro method for the construction of the Bernstein-Bézier algebraic hypersurface piece on a simplex with a prescribed topology is shown in the following proposition (see [18]).Proposition 3 (see [18]).
LetP,P1,…,Pm,ν,f1,…,fm, and ft be as above (ft is a patchwork of BB-polynomials f1,…,fm by ν). If BB-polynomials f1,…,fm are nondegenerate, then there exists t0>0 such that for any t∈(0,t0] the chart of BB-polynomial ft is obtained by patchworking charts of BB-polynomials f1,…,fm.


## 3. The Chart of the Piecewise Polynomial

In this section, the chart of the piecewise polynomial is defined, and some properties of the chart of the Bernstein-Bézier polynomial and the chart of the piecewise polynomial are discussed.Theorem 4.
Letf be a BB-polynomial with Newton polyhedron P and let Γ be a face of P. Then,(1)
CtS(f)⋂‍Γ=CtS(fΓ);(2)
iffΓ is nondegenerate with respect to Γ (which is the case when, for example, f is nondegenerate with respect to P), then, the CtS(f) intersects Γ transversally.Proof.
Suppose that the BB-polynomialf is defined in (4) and
(9)g(x)=∑|λ|=n‍n!λ1!⋯λk+1!bλx2λ2⋯xk+1λk+1.
Define the mappingψ:S→T(k,n)={(x2,…,xk+1)∈(ℝ+)k|x2+⋯+xk+1≤n} by
(10)ψ(τ1,τ2,…,τk+1)=(nτ2,…,nτk+1),0000000000000(τ1,τ2,…,τk+1)∈S.
According to [18, Lemma  3.2], then, ψ(P) is the Newton polyhedron N(g) of the polynomial g and ψ(Γ) is a face of N(g) if Γ is a face of P. Moreover, by [10, Remark  1.1], the chartCt+(g) of g (see [6–8, 18]) and the truncation gψ(Γ) on the face ψ(Γ) have the following properties:(a)
Ct+(g)⋂‍ψ(Γ)=Ct+(gψ(Γ));(b)
thechartCt+(g) intersects ψ(Γ) transversally if gψ(Γ) is nondegenerate with respect to ψ(Γ).
On the other hand, by [18, Lemma 3.2 and Theorem  3.5], we can get the following equalities:
(11)ψ-1(Ct+(g)⋂‍ψ(Γ))=CtS(f)⋂‍Γ,ψ-1(Ct+(gψ(Γ)))=CtS(fΓ).
This, together with equality (10) and properties (a) and (b), shows that CtS(f)⋂‍Γ=CtS(fΓ) and the CtS(f) intersects Γ transversally if fΓ is nondegenerate with respect to Γ. This completes the proof.The following result is a generalization of Farin’s theorem (see [3, 15]) on high dimensional space.Proposition 5 (see [3, 15]).
Let(12)p(1)(τ)=∑|λ|=n‍bλ(1)Bλ,n(τ),p(2)(τ¯)=∑|λ|=n‍bλ(2)Bλ,n(τ¯)
be BB-polynomials of degree n that are defined on two adjacent k-dimensional simplices S1=[v1,v2,…,vk+1] and S2=[v1′,v2,…,vk+1], respectively. Then p(1)(τ) and p(2)(τ¯) are Cr smoothly connected on [v2,…,vk+1] if and only if for all ρ∈{0,1,…,r} and λ2,…,λk+1∈ℤ+ with ∑j=2k+1λj=n-ρ,
(13)bλρ(2)=∑|ϵ|=ρ‍bϵ+λ0(1)Bϵ,ρ(τ′),
where λ0=(0,λ2,…,λk+1), λρ=(ρ,λ2,…,λk+1), and ϵ=(ϵ1,ϵ2,…,ϵk+1). Here, τ′ denotes the barycentric coordinate of v1′ with respect to the simplex S1.Theorem 6.
Suppose thatp(1)(τ) and p(2)(τ¯) defined as above are Cr(r≥0) smoothly connected on S1⋂‍S2=[v2,…,vk+1]. Then,(1)
N(p(1))⋂‍(S1⋂‍S2)=N(p(2))⋂‍(S1⋂‍S2), and Γ∶=N(p(1))⋂‍(S1⋂‍S2) is a face of N(p(1)) and N(p(2)) if N(p(1))⋂‍(S1⋂‍S2) is nonempty, where N(p(1)) and N(p(2)) are Newton polyhedra of BB-polynomials p(1) and p(2), respectively;(2)
CtS1(p(1))⋂‍Γ=CtS2(p(2))⋂‍Γ for Γ above.Proof.
By assumption and takingρ=0 in equality (13), we have that bλ0(1)=bλ0(2) for all λ2,…,λk+1∈ℤ+ with ∑j=2k+1λj=n. Thus, μ(p(1))⋂‍(S1⋂‍S2)=μ(p(2))⋂‍(S1⋂‍S2), where μ is defined in (6). This implies that N(p(1))⋂‍(S1⋂‍S2)=N(p(2))⋂‍(S1⋂‍S2), and so Γ=N(p(1))⋂‍(S1⋂‍S2) is a face of N(p(1)) and N(p(2)) if N(p(1))⋂‍(S1⋂‍S2) is nonempty.
According to the argument above, it is easy to see thatΓ-truncations (p(1))Γ, (p(2))Γ of p(1) and p(2) satisfy the equality (p(1))Γ=(p(2))Γ. Thus, we can get that
(14)CtS1((p(1))Γ)=CtS2((p(2))Γ)
by the definition of the chart.
On the other hand, it follows from Theorem4 that
(15)CtS1(p(1))⋂‍Γ=CtS1((p(1))Γ),CtS2(p(2))⋂‍Γ=CtS2((p(2))Γ).
Therefore,
(16)CtS1(p(1))⋂‍Γ=CtS2(p(2))⋂‍Γ
by equality (14). This completes the proof.LetΔ be a pure, hereditary simplicial complex with k-cells (i.e., k-dimensional simplex) S1,…,Sq. Let f∈Cnr(Δ) and each polynomial f|Sα be expressed in the B-form
(17)f|Sα(τ)=∑|λ|=n‍bλ[α]Bλ,n(τ),α=1,2,…,q.
The piecewise polynomial f defined as above is called Cr piecewise BB-polynomial.Theorem 7.
Letf∈Cnr(Δ) be a piecewise BB-polynomial defined in (17). Then, for each adjacent pair Sα, Sβ of k-cells in Δ,(1)
N(f|Sα)⋂‍(Sα⋂‍Sβ)=N(f|Sβ)⋂‍(Sα⋂‍Sβ), and Γαβ∶=N(f|Sα)⋂‍(Sα⋂‍Sβ) is a face of N(f|Sα) and N(f|Sβ) if N(f|Sα)⋂‍(Sα⋂‍Sβ) is nonempty, where N(f|Sα) and N(f|Sβ) are Newton polyhedra of BB-polynomials f|Sα and f|Sβ, respectively;(2)
CtSα(f|Sα)⋂‍Γαβ=CtSβ(f|Sβ)⋂‍Γαβ for Γαβ above.Proof.
We can get the conclusion from Theorem6 immediately.According to Theorems4–7, we can define the chart of a piecewise BB-polynomial.Definition 8.
Letf∈Cnr(Δ) be a piecewise BB-polynomial defined in (17), and let N(f|Sα) be the Newton polyhedron of the BB-polynomial f|Sα. The chart Ct(f) of f is the closure of ⋃α=1q‍ϕμ(f|Sα)({τ∈Int(Sα),f|Sα(τ)=0}), where μ(f|Sα)={(λ1/n,…,λk+1/n)|λ∈ℤ+(k,n),bλ[α]≠0} and ϕμ(f|Sα):Int(Sα)→Int(N(f|Sα)) is the moment map associated with μ(f|Sα) relative to the simplex Sα.Obviously, we can get the following conclusion from Theorems4–7 and the definition above immediately.Theorem 9.
Letf∈Cnr(Δ) be a piecewise BB-polynomial defined in (17). Then, the chartCt(f) of f is ⋃α=1q‍CtSα(f|Sα).


## 4. The Construction of RealCr Piecewise Algebraic Hypersurfaces

In this section, we propose a new method for the construction of real piecewise algebraic hypersurfaces of a given degree with certain smoothness and prescribed topology. The method is primarily based on the work of Lai et al. in [18] and the smooth blending theory.LetΔ be a pure, hereditary simplicial complex with k-cells (i.e., k-dimensional simplex) S1,…,Sq.For each adjacent pairSα=[v(α,1),v(α,2),…,v(α,k+1)], Sβ=[v(β,1),v(β,2),…,v(β,k+1)] of k-cells in Δ, assume that Sα∩Sβ=[v(α,1),…,v(α,i-1),v(α,i)^,v(α,i+1),…,v(α,k+1)]=[v(β,1),…,v(β,j-1),v(β,j)^,v(β,j+1),…,v(β,k+1)], where the hat means that the corresponding vertex is omitted. Obviously, there exist two arrangements l1,l2,…,lk+1 and d1,d2,…,dk+1 of numbers 1,2,…,k+1 such that v(α,l1)=v(α,i),v(β,d1)=v(β,j) and v(α,lw)=v(β,dw) for w=2,…,k+1.Now, we define two one-to-one transformationsηαβ, ηβα:(ℝ+)k+1→(ℝ+)k+1, by ηαβ(λ)=(λl1,λl2,…,λlk+1), and ηβα(λ)=(λd1,λd2,…,λdk+1), respectively, where λ=(λ1,λ2,…,λk+1)∈(ℝ+)k+1.For any givenα∈{1,…,q}, let f(α,1),…,f(α,mα) be BB-polynomials over the simplex Sα=[v(α,1),v(α,2),…,v(α,k+1)] with N(f(α,ζ))=P(α,ζ), ζ=1,…,mα, and let
(18)ft[α](τ)=∑|λ|=n‍bλ[α]Bλ,n(τ)tν[α](λ/n)
be a patchworking of BB-polynomials f(α,1),…,f(α,mα) by ν[α].Define the piecewise BB-polynomialft on the domain Ω by
(19)ft|Sα(τ)=ft[α](τ),α=1,2,…,q.Theorem 10.
Suppose thatf(1,1),…,f(1,m1),f(2,1),…,f(q,mq) are non-degenerate, that ft[α] is a patchworking of BB-polynomials f(α,1),…,f(α,mα) by ν[α], α=1,…,q, and that Sα, Sβ, v(α,i), v(β,j), ηαβ, ηβα, P(α,ζ), ft[α], and ft are defined as above, and Int(P(α,ζ))⋂‍Int(P(α,ς))=∅ for ζ≠ς. If for each adjacent pair Sα, Sβ of k-cells in Δ and all ρ∈{0,1,…,r} and λ=(λ1,λ2,…,λk+1)∈ℤ+(k,n) with ∑i=2k+1λi=n-ρ,
(20)b(ηβα-1(λρ))[β]tν[β](ηβα-1(λρ)/n)0000=∑|ϵ|=ρ‍bηαβ-1(ϵ+λ0)[α]Bϵ,ρ(τβα′)tν[α](ηαβ-1(ϵ+λ0)/n),t∈ℝ+*,
where λρ=(ρ,λ2,…,λk+1) and τβα′ denotes the barycentric coordinate of v(β,j) with respect to the simplex Sα=[v(α,l1),v(α,l2),…,v(α,lk+1)], then,(1)
ft∈Cnr(Δ);(2)
if for any BB-polynomialf(α,i) with N(f(α,i))⋂‍(Sα⋂‍Sβ)≠∅, i∈{1,…,mα}, there exists a BB-polynomial f(β,j) in {f(β,1),…,f(β,mβ)} such that N(f(β,j))⋂‍(Sα⋂‍Sβ)=N(f(α,i))⋂‍(Sα⋂‍Sβ), then there is t0>0 such that for any t∈(0,t0], the chart Ct(ft) of piecewise BB-polynomial ft, up to isotopy, is ⋃α=1q‍⋃i=1mα‍CtSα(f(α,i)).Proof.
Letτ=(τ1,…,τk+1) and χ=(χ1,χ2,…,χk+1) be the barycentric coordinates of point p∈Sβ with respect to Sβ=[v(β,1),v(β,2),…,v(β,k+1)] and Sβ=[v(β,d1),v(β,d2),…,v(β,dk+1)], respectively. Then,
(21)χ=(χ1,χ2,…,χk+1)=ηβα(τ)=(τd1,τd2,…,τdk+1)
by the definition of the transformation ηβα, and so
(22)ft[β](p)=ft[β](τ)=∑|λ|=n‍bλ[β]n!λ1!λ2!⋯λk+1!0000×τ1λ1τ2λ2⋯τk+1λk+1tν[β](λ/n)=∑|λ|=n‍bλ[β]n!λd1!λd1!⋯λdk+1!0000×(χ1)λd1(χ2)λd2⋯(χk+1)λdk+1tν[β](λ/n).
Set μ=(μ1,μ2,…,μk+1)∶=ηβα(λ). Then, λ=ηβα-1(μ) and (μ1,μ2,…,μk+1)=(λd1,λd2,…,λdk+1). Therefore, it follows from equality (22) that
(23)ft[β](p)=∑|μ|=n‍bηβα-1(μ)[β]tν[β](ηβα-1(μ)/n)0000×n!μ1!μ1!⋯μk+1!(χ1)μ1(χ2)μ2⋯(χk+1)μk+1.
This shows that the polynomial ft[β] over Sβ=[v(β,d1),v(β,d2),…,v(β,dk+1)] can be represented as follows:
(24)ft[β](χ)=∑|μ|=n‍bηβα-1(μ)[β]tν[β](ηβα-1(μ)/n)Bμ,n(χ).
By a similar argument above, we have that the polynomialft[α] over Sα=[v(α,l1),…,v(α,lk+1)] can be represented as follows
(25)ft[α](χ¯)=∑|μ|=n‍bηαβ-1(μ)[α]tν[α](ηαβ-1(μ)/n)Bμ,n(χ¯),
where χ¯=(χ¯1,χ¯2,…,χ¯k+1) is the barycentric coordinate of point p∈Sα with respect to Sα=[v(α,l1),v(α,l2),…,v(α,lk+1)].
Sincev(α,lw)=v(β,dw) for w=2,…,k+1, Sα⋂‍Sβ=[v(α,l2),…,v(α,lk+1)]; thus, according to Proposition 5, ft∈Cnr(Δ) if and only if the BB-coefficients bηβα-1(μ)[β]tν[β](ηβα-1(μ)/n) of ft[β] and the BB-coefficients bηαβ-1(μ)[α]tν[α](ηαβ-1(μ)/n) of ft[α] satisfy equality (20). This proves property (1).
Below we will show that property(2) holds.
For each adjacent pairSα, Sβ of k-cells in Δ, set Γαβ=N(f(α,i))⋂‍(Sα⋂‍Sβ) when N(f(α,i))⋂‍(Sα⋂‍Sβ)≠∅; then, Γαβ is a face of N(f(α,i)) and N(f(β,j)) by the assumption that N(f(β,j))⋂‍(Sα⋂‍Sβ)=N(f(α,i))⋂‍(Sα⋂‍Sβ). Since ft[α] (resp., ft[β]) is a patchworking of BB-polynomials f(α,1),…,f(α,mα) (resp., f(β,1),…,f(β,mβ)) by ν[α] (resp., ν[β]) and Int(P(α,ζ))⋂‍Int(P(α,ς))=Int(P(β,ζ))⋂‍Int(P(β,ς))=∅ for ζ≠ς, then, for Γαβ-truncations (f(α,i))Γαβ, (f(β,j))Γαβ, (ft[α])Γαβ and (ft[β])Γαβ of f(α,i), f(β,j), ft[α], and ft[β], respectively, we have that
(26)(f(α,i))Γαβ=(ft[α])Γαβ|t=1,(f(β,j))Γαβ=(ft[β])Γαβ|t=1.
Sinceft[α] and ft[β] are Cr smoothly connected on Sα⋂‍Sβ, N(f(β,j))⋂‍(Sα⋂‍Sβ)=N(f(α,i))⋂‍(Sα⋂‍Sβ) and (20) implies that (ft[α])Γαβ=(ft[β])Γαβ. Thus, (f(α,i))Γαβ=(f(β,j))Γαβ by (26).
Using a similar argument in the proof of Theorem6, we have CtSα(f(α,i))⋂‍Γαβ=CtSβ(f(β,j))⋂‍Γαβ by (f(α,i))Γαβ=(f(β,j))Γαβ. This shows that
(27)⋃i=1mα‍CtSα(f(α,i))⋂‍(Sα⋂‍Sβ)0000=⋃j=1mβ‍CtSβ(f(β,j))⋂‍(Sα⋂‍Sβ).
On the other hand, sinceft[α] is a patchworking of BB-polynomials f(α,1),…,f(α,mα) by ν[α], we get from the assumption and Proposition 3 that there is t0[α]>0 such that for any t∈(0,t0[α]] the chart CtSα(ft[α]) of BB-polynomial ft[α] (i.e., ft|Sα), up to isotopy, is ⋃i=1mα‍CtSα(f(α,i)). This conclusion together with Theorems 7–9 and equality (27) implies that there is t0>0 such that for any t∈(0,t0] the chart Ct(ft) of piecewise BB-polynomial ft, up to isotopy, is ⋃α=1q‍⋃i=1mα‍CtSα(f(α,i)), where t0=min{t0[1],…,t0[q]}. This completes the proof.Assume thatP is a k-dimensional simplex with domain points in S as its vertices and that the BB-polynomial f is a real k+1-nomial (this means that the nonzero coefficients in f correspond to the only vertices of P). Denote by M(P) the set of middle points of edges of P. For any point v∈M(P) on an edge with endpoints (i1/n,…,ik+1/n), (j1/n,…,jk+1/n), we assign  that
(28)δ(v)=sign(b(i1,…,ik+1)·b(j1,…,jk+1)),
where b(i1,…,ik+1),b(j1,…,jk+1) are the corresponding BB-coefficients in f. Put
(29)M-(P)={v∈M(P)∣δ(v)=-1}.Theorem 11.
Suppose thatf(1,1),…,f(1,m1),…,f(q,1),…,f(q,mq),P(1,1),…,P(1,m1),…,P(q,1),…,P(q,mq),ft[α], and ft satisfy the conditions in Theorem 10 and that for each α∈{1,2,…,q}, f(α,1),…,f(α,mα) are real (k+1)-nomial and P(α,1),…,P(α,mα) with dimension k form a triangulation of N(ft[α]). Then, ft∈Cnr(Δ) and there exists a constant t0>0 such that for any t∈(0,t0] the chart Ct(f) of f, up to isotopy, is ⋃α=1q‍⋃i=1mα‍conv(M-(P(α,i))), where M-(P(α,i)) is defined as (29).Proof.
By assumption and [18, Theorem  4.4], we know that there is t0[α]>0 such that for any t∈(0,t0[α]] the chart CtSα(ft[α]) of BB-polynomial ft[α], up to isotopy, is ⋃i=1mα‍conv(M-(P(α,i))).The remainder of the proof can be completed by a similar approach in the proof of Theorem10.According to Theorems10 and 11, if we want to construct a Cr piecewise algebraic hypersurface f(τ)=0 with a prescribed complex topology and a degree on a partition Δ, we can just construct some Bernstein-Bézier algebraic hypersurfaces pieces f(α,1)(τ)=0,…,f(α,mα)(τ)=0 with simple topology and a convex function ν[α] on each k-cell Sα, where f(α,1),…,f(α,mα), and ν[α] satisfy the conditions of Theorem 10 or Theorem 11, α=1,2,…,q, and then, the BB-polynomial f[α](τ) is obtained by a patchworking of BB-polynomials f(α,1),…,f(α,mα) by ν[α]. Thus, we can get f(τ) by defining f|Sα(τ)∶=f[α](τ).



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*Source: 690341-2013-09-29.xml*