The Viro Method for Construction of 𝐶^𝑟 Piecewise Algebraic Hypersurfaces

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.

1. Introduction

Let Δ be a simplicial subdivision of a region Ω in 𝑅^𝑘. 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,…,𝑆_𝑚=𝑆^′ in Δ such that each 𝑆_𝑖 is 𝑘-dimensional, each 𝑆_𝑖 contains 𝜏, and 𝑆_𝑖 and 𝑆_𝑖+1 are adjacent for each 𝑖 (see [1, 2]).

Let Δ be a pure, hereditary 𝑘-dimensional simplicial complex in 𝑅^𝑘, let 𝑆₁,𝑆₂,…,𝑆_𝑞 be a given, fixed, ordering of the 𝑘-cells in Δ, and let Ω=⋃^𝑞_𝑖=1𝑆_𝑖. Now, we recall the definitions of 𝐶^𝑟(Δ) and 𝐶^𝑟_𝑛(Δ) (see [1, 2]).

Definition 1. For a nonnegative integer 𝑟 and a 𝑘-complex Δ, 𝐶^𝑟(Δ) is the set of 𝐶^𝑟 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,…,𝑥_𝑘]. 𝐶^𝑟_𝑛(Δ) is the subset of 𝑓∈𝐶^𝑟(Δ) such that the restriction of 𝑓 to each cell in Δ is a polynomial function of degree 𝑘 or less.

It is clear that 𝐶^𝑟(Δ) and 𝐶^𝑟_𝑛(Δ) are a Noether ring and a finite dimensional linear vector space, respectively, and are called a 𝐶^𝑟 spline ring and a multivariate spline space with degree 𝑛 and smoothness 𝑟, respectively. We call 𝒵𝑓∶=𝑥₁,…,𝑥_𝑘∈Ω∣𝑓𝑥₁,…,𝑥_𝑘=0 (1) a real 𝐶^𝑟piecewise algebraic hypersurface (see [1–3]), where 𝑓∈𝐶^𝑟(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 {𝑃_𝑖,𝑖∈𝐼} of a polytope 𝑃 and a collection {𝑓_𝑖,𝑖∈𝐼} of real nondegenerate polynomials 𝑓_𝑖 with Newton polyhedra 𝑃_𝑖 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(𝑓_𝑖).

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. Section 2 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 by ℝ₊ (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,…,𝑣_𝑘+1] will be a fixed 𝑘-dimensional simplex with vertices 𝑣₁,…,𝑣_𝑘+1.

It is well known (cf. [3, 15]) that for any point 𝑝∈𝑆, it can be expressed uniquely as 𝑝=^𝑘+1_𝑗=1𝜏_𝑗𝑣_𝑗, (2) where ∑^𝑘+1_𝑗=1𝜏_𝑗=1,  𝜏_𝑗≥0, 𝑗=1,2,…,𝑘+1, and 𝜏=(𝜏₁,…,𝜏_𝑘+1) is the barycentric coordinates of 𝑝 with respect to 𝑆 (we abusively confuse a point 𝑝=∑^𝑘+1_𝑗=1𝜏_𝑗𝑣_𝑗∈𝑆 with the corresponding barycentric coordinates 𝜏=(𝜏₁,…,𝜏_𝑘+1)).

Let ℤ_+(𝑘,𝑛)=𝜆=𝜆₁,…,𝜆_𝑘+1∈ℤ₊^𝑘+1∣|𝜆|=^𝑘+1_𝑖=1𝜆_𝑖=𝑛. (3)

We call a point 𝑝∈𝑆 domain point if 𝑝=(𝜆₁/𝑛,…,𝜆_𝑘+1/𝑛) with (𝜆₁,…,𝜆_𝑘+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: 𝑓(𝜏)=_|𝜆|=𝑛𝑏_𝜆𝐵_𝜆,𝑛(𝜏), (4) where 𝑏_𝜆∈ℝ, 𝜆=(𝜆₁,…,𝜆_𝑘+1)∈ℤ_+(𝑘,𝑛), 𝐵_𝜆,𝑛(𝜏)=𝑛!𝜆₁!⋯𝜆_𝑘+1!𝜏^𝜆1₁⋯𝜏^𝜆𝑘+1_𝑘+1 (5) 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 𝑏_𝜆 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-polynomial 𝑓 defined in (4), set 𝜇𝑓𝜆∶=¹𝑛𝜆,…,^𝑘+1𝑛∣𝜆∈ℤ_+(𝑘,𝑛),𝑏_𝜆≠0. (6) 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 set Γ⊂𝑆 and a BB-polynomial 𝑓=∑_|𝜆|=𝑛𝑏_𝜆𝐵_𝜆,𝑛(𝜏), denote BB-polynomial ∑_{|𝜆|=𝑛,𝜆∈Γ}𝑏_𝜆𝐵_𝜆,𝑛(𝜏) by 𝑓^Γ. It is called the Γ-truncation of 𝑓.

A BB-polynomial 𝑓 is called nondegenerate if 𝑓 and any truncation 𝑓^𝛿 on a proper face 𝛿 of 𝑁(𝑓) has a nonsingular zero set in (ℂ^∗)^𝑘+1.

Let 𝒥_𝑆 be a subset of the set of domain points in simplex 𝑆 and 𝑃_𝑆=conv(𝒥_𝑆). Define the moment map associated with 𝒥_𝑆 relative to the simplex 𝑆, 𝜙_𝒥𝑆∶Int(𝑆)→Int(𝑃_𝑆), by 𝜙_𝒥𝑆(𝜏)=∑_{(𝑖1/𝑛,…,𝑖𝑘+1/𝑛)∈𝒥𝑆}𝜏^𝑖1₁⋯𝜏^𝑖𝑘+1_𝑘+1𝑖₁/𝑛,…,𝑖_𝑘+1/𝑛∑_{(𝑖1/𝑛,…,𝑖𝑘+1/𝑛)∈𝒥𝑆}𝜏^𝑖1₁⋯𝜏^𝑖𝑘+1_𝑘+1, (7) where Int(𝑃_𝑆) is the complement in 𝑃_𝑆 of the union of all its proper faces.

Definition 2. Let 𝑓 be a BB-polynomial with a Newton polyhedron 𝑃. Then the closure of 𝜙_𝜇(𝑓)({𝜏∈Int(𝑆),𝑓(𝜏)=0}) in 𝑃 is called chart 𝐶𝑡_𝑆(𝑓) of 𝑓, where 𝜙_𝜇(𝑓)∶Int(𝑆)→Int(𝑃) is the moment map associated with 𝜇(𝑓) relative to the simplex 𝑆.

Let 𝑃,𝑃₁,…,𝑃_𝑚⊂𝑆 be polyhedra with 𝑃=⋃^𝑚_𝑖=1𝑃_𝑖 and Int(𝑃_𝑖)⋂Int(𝑃_𝑗)=∅ for 𝑖≠𝑗. Assume that 𝜈∶𝑃→ℝ is a continuous, piecewise linear, nonnegative convex function satisfying the following conditions:(1) all the restrictions 𝜈|_𝑃𝑖 are linear;(2) if the restriction of 𝜈 to an open set is linear, then this set is contained in one of the 𝑃_𝑖;(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,…,𝑃_𝑚}.

Let 𝑓₁,…,𝑓_𝑚 be BB-polynomials over ℝ in 𝑘+1 variables with 𝑁(𝑓_𝑖)=𝑃_𝑖. Let 𝑓^{𝑃𝑖⋂𝑃𝑗}_𝑖=𝑓^{𝑃𝑖⋂𝑃𝑗}_𝑗 for any 𝑖,𝑗. Then, there exists a unique BB-polynomial 𝑓 with 𝑁(𝑓)=𝑃=⋃^𝑚_𝑖=1𝑃_𝑖 and 𝑓^𝑃𝑖=𝑓_𝑖 for 𝑖=1,…,𝑚. If 𝑓(𝜏)=∑_|𝜆|=𝑛𝑏_𝜆𝐵_𝜆,𝑛(𝜏) and 𝜈 is a function convexifying {𝑃₁,…,𝑃_𝑚}, we put 𝑓_𝑡(𝜏)=_|𝜆|=𝑛𝑏_𝜆𝐵_𝜆,𝑛(𝜏)𝑡^{𝜈(𝜆/𝑛)}. (8) The BB-polynomials 𝑓_𝑡 are said to be obtained by 𝑝𝑎𝑡𝑐ℎ𝑤𝑜𝑟𝑘𝑖𝑛𝑔 BB-polynomials 𝑓₁,…,𝑓_𝑚 by 𝜈 or, briefly, 𝑓_𝑡 is a 𝑝𝑎𝑡𝑐ℎ𝑤𝑜𝑟𝑘 of BB-polynomials 𝑓₁,…,𝑓_𝑚 by 𝜈.

Let 𝑓₁,…,𝑓_𝑚 be BB-polynomials in 𝑘+1 variables with 𝑁(𝑓_𝑖)=𝑃_𝑖, and let Int(𝑃_𝑖)⋂Int(𝑃_𝑗)=∅ for 𝑖≠𝑗. A chart 𝐶𝑡_𝑆(𝑓) of a BB-polynomial 𝑓 with 𝑁(𝑓)=𝑃 is said to be obtained by 𝑝𝑎𝑡𝑐ℎ𝑤𝑜𝑟𝑘𝑖𝑛𝑔 charts of BB-polynomials 𝑓₁,…,𝑓_𝑚 and it is a 𝑝𝑎𝑡𝑐ℎ𝑤𝑜𝑟𝑘 of charts of BB-polynomials 𝑓₁,…,𝑓_𝑚 if 𝑃=⋃^𝑚_𝑖=1𝑃_𝑖 and the chart 𝐶𝑡_𝑆(𝑓) of 𝑓, up to isotopy, is ⋃^𝑚_𝑖=1𝐶𝑡_𝑆(𝑓_𝑖).

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]). Let 𝑃,𝑃₁,…,𝑃_𝑚,𝜈,𝑓₁,…,𝑓_𝑚, and 𝑓_𝑡 be as above (𝑓_𝑡 is a 𝑝𝑎𝑡𝑐ℎ𝑤𝑜𝑟𝑘 of BB-polynomials 𝑓₁,…,𝑓_𝑚 by 𝜈). If BB-polynomials 𝑓₁,…,𝑓_𝑚 are nondegenerate, then there exists 𝑡₀>0 such that for any 𝑡∈(0,𝑡₀] the chart of BB-polynomial 𝑓_𝑡 is obtained by patchworking charts of BB-polynomials 𝑓₁,…,𝑓_𝑚.

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. Let 𝑓 be a BB-polynomial with Newton polyhedron 𝑃 and let Γ be a face of 𝑃. Then,(1)𝐶𝑡_𝑆(𝑓)⋂Γ=𝐶𝑡_𝑆(𝑓^Γ);(2) if 𝑓^Γ is nondegenerate with respect to Γ (which is the case when, for example, 𝑓 is nondegenerate with respect to 𝑃), then, the 𝐶𝑡_𝑆(𝑓) intersects Γ transversally.

Proof. Suppose that the BB-polynomial 𝑓 is defined in (4) and 𝑔(𝑥)=_|𝜆|=𝑛𝑛!𝜆₁!⋯𝜆_𝑘+1!𝑏_𝜆𝑥^𝜆2₂⋯𝑥^𝜆𝑘+1_𝑘+1. (9)
Define the mapping 𝜓∶𝑆→𝑇_(𝑘,𝑛)={(𝑥₂,…,𝑥_𝑘+1)∈(ℝ₊)^𝑘|𝑥₂+⋯+𝑥_𝑘+1≤𝑛} by 𝜓𝜏₁,𝜏₂,…,𝜏_𝑘+1=𝑛𝜏₂,…,𝑛𝜏_𝑘+1,𝜏₁,𝜏₂,…,𝜏_𝑘+1∈𝑆. (10) 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 𝑔^𝜓(Γ) on the face 𝜓(Γ) have the following properties:(a)𝐶𝑡₊(𝑔)⋂𝜓(Γ)=𝐶𝑡₊(𝑔^𝜓(Γ));(b) the 𝑐ℎ𝑎𝑟𝑡𝐶𝑡₊(𝑔) intersects 𝜓(Γ) transversally if 𝑔^𝜓(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: 𝜓⁻¹𝐶𝑡₊𝑔𝜓(Γ)=𝐶𝑡_𝑆𝑓𝜓Γ,₋₁𝐶𝑡₊𝑔^𝜓(Γ)=𝐶𝑡_𝑆𝑓^Γ. (11) This, together with equality (10) and properties (a) and (b), shows that 𝐶𝑡_𝑆(𝑓)⋂Γ=𝐶𝑡_𝑆(𝑓^Γ) and the 𝐶𝑡_𝑆(𝑓) intersects Γ transversally if 𝑓^Γ 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 𝑝⁽¹⁾(𝜏)=_|𝜆|=𝑛𝑏_𝜆⁽¹⁾𝐵_𝜆,𝑛(𝜏),𝑝⁽²⁾(𝜏)=_|𝜆|=𝑛𝑏_𝜆⁽²⁾𝐵_𝜆,𝑛(𝜏) (12) be BB-polynomials of degree 𝑛 that are defined on two adjacent 𝑘-dimensional simplices 𝑆₁=[𝑣₁,𝑣₂,…,𝑣_𝑘+1] and 𝑆₂=[𝑣^′₁,𝑣₂,…,𝑣_𝑘+1], respectively. Then 𝑝⁽¹⁾(𝜏) and 𝑝⁽²⁾(𝜏) are 𝐶^𝑟 smoothly connected on [𝑣₂,…,𝑣_𝑘+1] if and only if for all 𝜌∈{0,1,…,𝑟} and 𝜆₂,…,𝜆_𝑘+1∈ℤ₊ with ∑^𝑘+1_𝑗=2𝜆_𝑗=𝑛−𝜌, 𝑏_𝜆⁽²⁾_𝜌=_|𝜖|=𝜌𝑏⁽¹⁾_𝜖+𝜆0𝐵_𝜖,𝜌𝜏^′, (13) where 𝜆⁰=(0,𝜆₂,…,𝜆_𝑘+1), 𝜆^𝜌=(𝜌,𝜆₂,…,𝜆_𝑘+1), and 𝜖=(𝜖₁,𝜖₂,…,𝜖_𝑘+1). Here, 𝜏^′ denotes the barycentric coordinate of 𝑣^′₁ with respect to the simplex 𝑆₁.

Theorem 6. Suppose that 𝑝⁽¹⁾(𝜏) and 𝑝⁽²⁾(𝜏) defined as above are 𝐶^𝑟(𝑟≥0) smoothly connected on 𝑆₁⋂𝑆₂=[𝑣₂,…,𝑣_𝑘+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)𝐶𝑡_𝑆1(𝑝⁽¹⁾)⋂Γ=𝐶𝑡_𝑆2(𝑝⁽²⁾)⋂Γ for Γ above.

Proof. By assumption and taking 𝜌=0 in equality (13), we have that 𝑏_𝜆⁽¹⁾₀=𝑏_𝜆⁽²⁾₀ for all 𝜆₂,…,𝜆_𝑘+1∈ℤ₊ with ∑^𝑘+1_𝑗=2𝜆_𝑗=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 that Γ-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 𝐶𝑡_𝑆1𝑝^(1)Γ=𝐶𝑡_𝑆2𝑝^(2)Γ (14) by the definition of the 𝑐ℎ𝑎𝑟𝑡.
On the other hand, it follows from Theorem 4 that 𝐶𝑡_𝑆1𝑝⁽¹⁾Γ=𝐶𝑡_𝑆1𝑝^(1)Γ,𝐶𝑡_𝑆2𝑝⁽²⁾Γ=𝐶𝑡_𝑆2𝑝^(2)Γ. (15) Therefore, 𝐶𝑡_𝑆1𝑝⁽¹⁾Γ=𝐶𝑡_𝑆2𝑝⁽²⁾Γ (16) by equality (14). This completes the proof.

Let Δ be a pure, hereditary simplicial complex with 𝑘-cells (i.e., 𝑘-dimensional simplex) 𝑆₁,…,𝑆_𝑞. Let 𝑓∈𝐶^𝑟_𝑛(Δ) and each polynomial 𝑓|_𝑆𝛼 be expressed in the B-form 𝑓_𝑆𝛼(𝜏)=_|𝜆|=𝑛𝑏_𝜆^[𝛼]𝐵_𝜆,𝑛(𝜏),𝛼=1,2,…,𝑞. (17) The piecewise polynomial 𝑓 defined as above is called 𝐶^𝑟 piecewise BB-polynomial.

Theorem 7. Let 𝑓∈𝐶^𝑟_𝑛(Δ) be a piecewise BB-polynomial defined in (17). Then, for each adjacent pair 𝑆_𝛼, 𝑆_𝛽 of 𝑘-cells in Δ,(1)𝑁(𝑓|_𝑆𝛼)⋂(𝑆_𝛼⋂𝑆_𝛽)=𝑁(𝑓|_𝑆𝛽)⋂(𝑆_𝛼⋂𝑆_𝛽), and Γ_𝛼𝛽∶=𝑁(𝑓|_𝑆𝛼)⋂(𝑆_𝛼⋂𝑆_𝛽) is a face of 𝑁(𝑓|_𝑆𝛼) and 𝑁(𝑓|_𝑆𝛽) if 𝑁(𝑓|_𝑆𝛼)⋂(𝑆_𝛼⋂𝑆_𝛽) is nonempty, where 𝑁(𝑓|_𝑆𝛼) and 𝑁(𝑓|_𝑆𝛽) are Newton polyhedra of BB-polynomials 𝑓|_𝑆𝛼 and 𝑓|_𝑆𝛽, respectively;(2)𝐶𝑡_𝑆𝛼(𝑓|_𝑆𝛼)⋂Γ_𝛼𝛽=𝐶𝑡_𝑆𝛽(𝑓|_𝑆𝛽)⋂Γ_𝛼𝛽 for Γ_𝛼𝛽 above.

Proof. We can get the conclusion from Theorem 6 immediately.

According to Theorems 4–7, we can define the 𝑐ℎ𝑎𝑟𝑡 of a piecewise BB-polynomial.

Definition 8. Let 𝑓∈𝐶^𝑟_𝑛(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 ⋃^𝑞_𝛼=1𝜙_{𝜇(𝑓|𝑆𝛼)}({𝜏∈Int(𝑆_𝛼),𝑓|_𝑆𝛼(𝜏)=0}), where 𝜇(𝑓|_𝑆𝛼)={(𝜆₁/𝑛,…,𝜆_𝑘+1/𝑛)|𝜆∈ℤ_+(𝑘,𝑛),𝑏_𝜆^[𝛼]≠0} and 𝜙_{𝜇(𝑓|𝑆𝛼)}∶Int(𝑆_𝛼)→Int(𝑁(𝑓|_𝑆𝛼)) is the moment map associated with 𝜇(𝑓|_𝑆𝛼) relative to the simplex 𝑆_𝛼.

Obviously, we can get the following conclusion from Theorems 4–7 and the definition above immediately.

Theorem 9. Let 𝑓∈𝐶^𝑟_𝑛(Δ) be a piecewise BB-polynomial defined in (17). Then, the 𝑐ℎ𝑎𝑟𝑡𝐶𝑡(𝑓) of 𝑓 is ⋃^𝑞_𝛼=1𝐶𝑡_𝑆𝛼(𝑓|_𝑆𝛼).

4. The Construction of Real 𝐶^𝑟 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.

Let Δ be a pure, hereditary simplicial complex with 𝑘-cells (i.e., 𝑘-dimensional simplex) 𝑆₁,…,𝑆_𝑞.

For each adjacent pair 𝑆_𝛼=[𝑣_(𝛼,1),𝑣_(𝛼,2),…,𝑣_{(𝛼,𝑘+1)}], 𝑆_𝛽=[𝑣_(𝛽,1),𝑣_(𝛽,2),…,𝑣_{(𝛽,𝑘+1)}] of 𝑘-cells in Δ, assume that 𝑆_𝛼∩𝑆_𝛽=[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,…,𝑙_𝑘+1 and 𝑑₁,𝑑₂,…,𝑑_𝑘+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 transformations 𝜂_𝛼𝛽, 𝜂_𝛽𝛼∶(ℝ₊)^𝑘+1→(ℝ₊)^𝑘+1, by 𝜂_𝛼𝛽(𝜆)=(𝜆_𝑙1,𝜆_𝑙2,…,𝜆_𝑙𝑘+1), and 𝜂_𝛽𝛼(𝜆)=(𝜆_𝑑1,𝜆_𝑑2,…,𝜆_𝑑𝑘+1), respectively, where 𝜆=(𝜆₁,𝜆₂,…,𝜆_𝑘+1)∈(ℝ₊)^𝑘+1.

For any given 𝛼∈{1,…,𝑞}, let 𝑓^(𝛼,1),…,𝑓^{(𝛼,𝑚𝛼)} be BB-polynomials over the simplex 𝑆_𝛼=[𝑣_(𝛼,1),𝑣_(𝛼,2),…,𝑣_{(𝛼,𝑘+1)}] with 𝑁(𝑓^(𝛼,𝜁))=𝑃_(𝛼,𝜁), 𝜁=1,…,𝑚_𝛼, and let 𝑓_𝑡^[𝛼](𝜏)=_|𝜆|=𝑛𝑏_𝜆^[𝛼]𝐵_𝜆,𝑛(𝜏)𝑡^{𝜈[𝛼](𝜆/𝑛)} (18) be a 𝑝𝑎𝑡𝑐ℎ𝑤𝑜𝑟𝑘𝑖𝑛𝑔 of BB-polynomials 𝑓^(𝛼,1),…,𝑓^{(𝛼,𝑚𝛼)} by 𝜈^[𝛼].

Define the piecewise BB-polynomial 𝑓_𝑡 on the domain Ω by 𝑓_𝑡𝑆𝛼(𝜏)=𝑓_𝑡^[𝛼](𝜏),𝛼=1,2,…,𝑞. (19)

Theorem 10. Suppose that 𝑓^(1,1),…,𝑓^(1,𝑚1),𝑓^(2,1),…,𝑓^{(𝑞,𝑚𝑞)} are non-degenerate, that 𝑓_𝑡^[𝛼] is a 𝑝𝑎𝑡𝑐ℎ𝑤𝑜𝑟𝑘𝑖𝑛𝑔 of BB-polynomials 𝑓^(𝛼,1),…,𝑓^{(𝛼,𝑚𝛼)} by 𝜈^[𝛼], 𝛼=1,…,𝑞, and that 𝑆_𝛼, 𝑆_𝛽, 𝑣_(𝛼,𝑖), 𝑣_(𝛽,𝑗), 𝜂_𝛼𝛽, 𝜂_𝛽𝛼, 𝑃_(𝛼,𝜁), 𝑓_𝑡^[𝛼], and 𝑓_𝑡 are defined as above, and Int(𝑃_(𝛼,𝜁))⋂Int(𝑃_(𝛼,𝜍))=∅ for 𝜁≠𝜍. If for each adjacent pair 𝑆_𝛼, 𝑆_𝛽 of 𝑘-cells in Δ and all 𝜌∈{0,1,…,𝑟} and 𝜆=(𝜆₁,𝜆₂,…,𝜆_𝑘+1)∈ℤ_+(𝑘,𝑛) with ∑^𝑘+1_𝑖=2𝜆_𝑖=𝑛−𝜌, 𝑏_𝜂^[𝛽]_{−1𝛽𝛼𝜆𝜌}𝑡^{𝜈[𝛽](𝜂−1𝛽𝛼(𝜆𝜌)/𝑛)}=_|𝜖|=𝜌𝑏_𝜂^[𝛼]_{−1𝛼𝛽𝜖+𝜆0}𝐵_𝜖,𝜌𝜏^′_𝛽𝛼𝑡^{𝜈[𝛼](𝜂−1𝛼𝛽(𝜖+𝜆0)/𝑛)},𝑡∈ℝ^∗₊, (20) where 𝜆^𝜌=(𝜌,𝜆₂,…,𝜆_𝑘+1) and 𝜏^′_𝛽𝛼 denotes the barycentric coordinate of 𝑣_(𝛽,𝑗) with respect to the simplex 𝑆_𝛼=[𝑣_(𝛼,𝑙1),𝑣_(𝛼,𝑙2),…,𝑣_{(𝛼,𝑙𝑘+1)}], then,(1)𝑓_𝑡∈𝐶^𝑟_𝑛(Δ);(2) if for any BB-polynomial 𝑓^(𝛼,𝑖) with 𝑁(𝑓^(𝛼,𝑖))⋂(𝑆_𝛼⋂𝑆_𝛽)≠∅, 𝑖∈{1,…,𝑚_𝛼}, there exists a BB-polynomial 𝑓^(𝛽,𝑗) in {𝑓^(𝛽,1),…,𝑓^{(𝛽,𝑚𝛽)}} such that 𝑁(𝑓^(𝛽,𝑗))⋂(𝑆_𝛼⋂𝑆_𝛽)=𝑁(𝑓^(𝛼,𝑖))⋂(𝑆_𝛼⋂𝑆_𝛽), then there is 𝑡₀>0 such that for any 𝑡∈(0,𝑡₀], the chart 𝐶𝑡(𝑓_𝑡) of piecewise BB-polynomial 𝑓_𝑡, up to isotopy, is ⋃^𝑞_𝛼=1⋃^𝑚𝛼_𝑖=1𝐶𝑡_𝑆𝛼(𝑓^(𝛼,𝑖)).

Proof. Let 𝜏=(𝜏₁,…,𝜏_𝑘+1) and 𝜒=(𝜒₁,𝜒₂,…,𝜒_𝑘+1) be the barycentric coordinates of point 𝑝∈𝑆_𝛽 with respect to 𝑆_𝛽=[𝑣_(𝛽,1),𝑣_(𝛽,2),…,𝑣_{(𝛽,𝑘+1)}] and 𝑆_𝛽=[𝑣_(𝛽,𝑑1),𝑣_(𝛽,𝑑2),…,𝑣_{(𝛽,𝑑𝑘+1)}], respectively. Then, 𝜒=𝜒₁,𝜒₂,…,𝜒_𝑘+1=𝜂_𝛽𝛼(𝜏)=𝜏_𝑑1,𝜏_𝑑2,…,𝜏_𝑑𝑘+1 (21) by the definition of the transformation 𝜂_𝛽𝛼, and so 𝑓_𝑡^[𝛽]𝑝=𝑓_𝑡^[𝛽](𝜏)=_|𝜆|=𝑛𝑏_𝜆^[𝛽]𝑛!𝜆₁!𝜆₂!⋯𝜆_𝑘+1!×𝜏^𝜆1₁𝜏^𝜆2₂⋯𝜏^𝜆𝑘+1_𝑘+1𝑡^{𝜈[𝛽](𝜆/𝑛)}=_|𝜆|=𝑛𝑏_𝜆^[𝛽]𝑛!𝜆_𝑑1!𝜆_𝑑1!⋯𝜆_𝑑𝑘+1!×𝜒₁^𝜆𝑑1𝜒₂^𝜆𝑑2⋯𝜒_𝑘+1^{𝜆𝑑𝑘+1}𝑡^{𝜈[𝛽](𝜆/𝑛)}. (22) Set 𝜇=(𝜇₁,𝜇₂,…,𝜇_𝑘+1)∶=𝜂_𝛽𝛼(𝜆). Then, 𝜆=𝜂⁻¹_𝛽𝛼(𝜇) and (𝜇₁,𝜇₂,…,𝜇_𝑘+1)=(𝜆_𝑑1,𝜆_𝑑2,…,𝜆_𝑑𝑘+1). Therefore, it follows from equality (22) that 𝑓_𝑡^[𝛽]𝑝=_𝜇=𝑛𝑏_𝜂^[𝛽]_{−1𝛽𝛼𝜇}𝑡^{𝜈[𝛽](𝜂−1𝛽𝛼(𝜇)/𝑛)}×𝑛!𝜇₁!𝜇₁!⋯𝜇_𝑘+1!𝜒₁^𝜇1𝜒₂^𝜇2⋯𝜒_𝑘+1^𝜇𝑘+1. (23) This shows that the polynomial 𝑓_𝑡^[𝛽] over 𝑆_𝛽=[𝑣_(𝛽,𝑑1),𝑣_(𝛽,𝑑2),…,𝑣_{(𝛽,𝑑𝑘+1)}] can be represented as follows: 𝑓_𝑡^[𝛽]𝜒=_𝜇=𝑛𝑏_𝜂^[𝛽]_{−1𝛽𝛼(𝜇)}𝑡^{𝜈[𝛽](𝜂−1𝛽𝛼(𝜇)/𝑛)}𝐵_𝜇,𝑛𝜒. (24)
By a similar argument above, we have that the polynomial 𝑓_𝑡^[𝛼] over 𝑆_𝛼=[𝑣_(𝛼,𝑙1),…,𝑣_{(𝛼,𝑙𝑘+1)}] can be represented as follows 𝑓_𝑡^[𝛼]𝜒=_𝜇=𝑛𝑏_𝜂^[𝛼]_{−1𝛼𝛽(𝜇)}𝑡^{𝜈[𝛼](𝜂−1𝛼𝛽(𝜇)/𝑛)}𝐵_𝜇,𝑛𝜒, (25) where 𝜒=(𝜒₁,𝜒₂,…,𝜒_𝑘+1) is the barycentric coordinate of point 𝑝∈𝑆_𝛼 with respect to 𝑆_𝛼=[𝑣_(𝛼,𝑙1),𝑣_(𝛼,𝑙2),…,𝑣_{(𝛼,𝑙𝑘+1)}].
Since 𝑣_{(𝛼,𝑙𝑤)}=𝑣_{(𝛽,𝑑𝑤)} for 𝑤=2,…,𝑘+1, 𝑆_𝛼⋂𝑆_𝛽=[𝑣_(𝛼,𝑙2),…,𝑣_{(𝛼,𝑙𝑘+1)}]; thus, according to Proposition 5, 𝑓_𝑡∈𝐶^𝑟_𝑛(Δ) if and only if the BB-coefficients 𝑏_𝜂^[𝛽]_{−1𝛽𝛼(𝜇)}𝑡^{𝜈[𝛽](𝜂−1𝛽𝛼(𝜇)/𝑛)} of 𝑓_𝑡^[𝛽] and the BB-coefficients 𝑏_𝜂^[𝛼]_{−1𝛼𝛽(𝜇)}𝑡^{𝜈[𝛼](𝜂−1𝛼𝛽(𝜇)/𝑛)} of 𝑓_𝑡^[𝛼] satisfy equality (20). This proves property (1).
Below we will show that property (2) holds.
For each adjacent pair 𝑆_𝛼, 𝑆_𝛽 of 𝑘-cells in Δ, set Γ_𝛼𝛽=𝑁(𝑓^(𝛼,𝑖))⋂(𝑆_𝛼⋂𝑆_𝛽) when 𝑁(𝑓^(𝛼,𝑖))⋂(𝑆_𝛼⋂𝑆_𝛽)≠∅; then, Γ_𝛼𝛽 is a face of 𝑁(𝑓^(𝛼,𝑖)) and 𝑁(𝑓^(𝛽,𝑗)) by the assumption that 𝑁(𝑓^(𝛽,𝑗))⋂(𝑆_𝛼⋂𝑆_𝛽)=𝑁(𝑓^(𝛼,𝑖))⋂(𝑆_𝛼⋂𝑆_𝛽). Since 𝑓_𝑡^[𝛼] (resp., 𝑓_𝑡^[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 Γ_𝛼𝛽-truncations (𝑓^(𝛼,𝑖))^Γ𝛼𝛽, (𝑓^(𝛽,𝑗))^Γ𝛼𝛽, (𝑓_𝑡^[𝛼])^Γ𝛼𝛽 and (𝑓_𝑡^[𝛽])^Γ𝛼𝛽 of 𝑓^(𝛼,𝑖), 𝑓^(𝛽,𝑗), 𝑓_𝑡^[𝛼], and 𝑓_𝑡^[𝛽], respectively, we have that 𝑓^{(𝛼,𝑖)Γ𝛼𝛽}=𝑓_𝑡^{[𝛼]Γ𝛼𝛽}_𝑡=1,𝑓^{(𝛽,𝑗)Γ𝛼𝛽}=𝑓_𝑡^{[𝛽]Γ𝛼𝛽}_𝑡=1. (26)
Since 𝑓_𝑡^[𝛼] and 𝑓_𝑡^[𝛽] are 𝐶^𝑟 smoothly connected on 𝑆_𝛼⋂𝑆_𝛽, 𝑁(𝑓^(𝛽,𝑗))⋂(𝑆_𝛼⋂𝑆_𝛽)=𝑁(𝑓^(𝛼,𝑖))⋂(𝑆_𝛼⋂𝑆_𝛽) and (20) implies that (𝑓_𝑡^[𝛼])^Γ𝛼𝛽=(𝑓_𝑡^[𝛽])^Γ𝛼𝛽. Thus, (𝑓^(𝛼,𝑖))^Γ𝛼𝛽=(𝑓^(𝛽,𝑗))^Γ𝛼𝛽 by (26).
Using a similar argument in the proof of Theorem 6, we have 𝐶𝑡_𝑆𝛼(𝑓^(𝛼,𝑖))⋂Γ_𝛼𝛽=𝐶𝑡_𝑆𝛽(𝑓^(𝛽,𝑗))⋂Γ_𝛼𝛽 by (𝑓^(𝛼,𝑖))^Γ𝛼𝛽=(𝑓^(𝛽,𝑗))^Γ𝛼𝛽. This shows that ^𝑚𝛼_𝑖=1𝐶𝑡_𝑆𝛼𝑓^(𝛼,𝑖)𝑆_𝛼𝑆_𝛽=^𝑚𝛽_𝑗=1𝐶𝑡_𝑆𝛽𝑓^(𝛽,𝑗)𝑆_𝛼𝑆_𝛽. (27)
On the other hand, since 𝑓_𝑡^[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 𝐶𝑡_𝑆𝛼(𝑓_𝑡^[𝛼]) of BB-polynomial 𝑓_𝑡^[𝛼] (i.e., 𝑓_𝑡|_𝑆𝛼), up to isotopy, is ⋃^𝑚𝛼_𝑖=1𝐶𝑡_𝑆𝛼(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(𝑓_𝑡) of piecewise BB-polynomial 𝑓_𝑡, up to isotopy, is ⋃^𝑞_𝛼=1⋃^𝑚𝛼_𝑖=1𝐶𝑡_𝑆𝛼(𝑓^(𝛼,𝑖)), where 𝑡₀=min{𝑡₀^[1],…,𝑡₀^[𝑞]}. This completes the proof.

Assume that 𝑃 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,…,𝑖_𝑘+1/𝑛), (𝑗₁/𝑛,…,𝑗_𝑘+1/𝑛), we assign that 𝛿(𝑣)=sign𝑏_{(𝑖1,…,𝑖𝑘+1)}⋅𝑏_{(𝑗1,…,𝑗𝑘+1)}, (28) where 𝑏_{(𝑖1,…,𝑖𝑘+1)},𝑏_{(𝑗1,…,𝑗𝑘+1)} are the corresponding BB-coefficients in 𝑓. Put 𝑀₋(𝑃)={𝑣∈𝑀(𝑃)∣𝛿(𝑣)=−1}. (29)

Theorem 11. Suppose that 𝑓^(1,1),…,𝑓^(1,𝑚1),…,𝑓^(𝑞,1),…,𝑓^{(𝑞,𝑚𝑞)},𝑃_(1,1),…,𝑃_(1,𝑚1),…,𝑃_(𝑞,1),…,𝑃_{(𝑞,𝑚𝑞)},𝑓_𝑡^[𝛼], and 𝑓_𝑡 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(𝑓_𝑡^[𝛼]). Then, 𝑓_𝑡∈𝐶^𝑟_𝑛(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 ⋃^𝑞_𝛼=1⋃^𝑚𝛼_𝑖=1conv(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 𝐶𝑡_𝑆𝛼(𝑓_𝑡^[𝛼]) of BB-polynomial 𝑓_𝑡^[𝛼], up to isotopy, is ⋃^𝑚𝛼_𝑖=1conv(𝑀₋(𝑃_(𝛼,𝑖))).

The remainder of the proof can be completed by a similar approach in the proof of Theorem 10.

According to Theorems 10 and 11, if we want to construct a 𝐶^𝑟 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 𝑆_𝛼, 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).

Conflict of Interests

The authors do not have any possible conflict of interests in this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 11271328, 11071031, U0935004, U1135003, 11290143, 11101366, 11226329, and 61272307), the Zhejiang Provincial Natural Science Foundation (nos. Y7080068, LQ13A010004), and the Foundation of Department of Education of Zhejiang Province (no. Y201223556).