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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 690341, 7 pages
The Viro Method for Construction of Piecewise Algebraic Hypersurfaces
1Department of Information and Computing Science, Zhejiang Gongshang University, Hangzhou 310018, China
2Institute of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
Received 30 July 2013; Accepted 22 August 2013
Academic Editor: Lawrence Narici
Copyright © 2013 Yisheng Lai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
Let be a simplicial subdivision of a region in . is called a pure simplicial complex of dimension and can be described as a finite collection of simplices such that the faces of each element of are elements of , and the intersection of any two elements of is an element of , and every maximal element of (with respect to inclusion) is a -dimensional simplex. We will sometimes refer to the -dimensional elements of as -cells and the simplicial subdivision as a -complex. If two -dimensional simplices in meet in a face of dimension , we say they are adjacent. is said to be hereditary if for every (including the empty set) any two -dimensional simplices of that contain can be connected by a sequence in such that each is -dimensional, each contains , and and are adjacent for each (see [1, 2]).
Definition 1. For a nonnegative integer and a -complex , is the set of functions on (i.e., functions such that all th order partial derivatives exist and are continuous on ) such that, for every including those of dimension , the restriction is a polynomial function . is the subset of such that the restriction of to each cell in is a polynomial function of degree or less.
It is clear that and are a Noether ring and a finite dimensional linear vector space, respectively, and are called a spline ring and a multivariate spline space with degree and smoothness , respectively. We call a real piecewise algebraic hypersurface (see [1–3]), where .
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 of a polytope and a collection of real nondegenerate polynomials with Newton polyhedra whose truncations on common faces of Newton polyhedra coincide. Then, a Viro polynomial with Newton polytope is defined, and Viro’s theorem asserts that the topology of the real hypersurface defined by can be recovered by gluing together pieces of the real hypersurfaces .
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  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  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  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 .
Throughout this paper, we denote by (resp., ) the set of real numbers such that (resp., ) and by the set of nonnegative integers. Let will be a fixed -dimensional simplex with vertices .
It is well known (cf. [3, 15]) that for any point , it can be expressed uniquely as where , , , and is the barycentric coordinates of with respect to (we abusively confuse a point with the corresponding barycentric coordinates ).
We call a point domain point if with . Further on, polyhedron relative to the simplex means a convex polyhedron in with domain points as its vertices.
It is well known (c.f. [3, 15]) that for any polynomial over with variables and degree at most , it can be represented in the Bernstein-Bézier form as follows: where , , are the Bernstein basis of degree relative to . The is called a Bernstein-Bézier polynomial or B-form of the polynomial relative to the simplex . We will refer to the polynomials in B-form as BB-polynomials and to their coefficients as BB-coefficients. The set of its zero points in is called a Bernstein-Bézier algebraic hypersurface piece (BB-algebraic hypersurface piece for short).
For the BB-polynomial defined in (4), set The convex hull of on , denoted by , is called Newton polyhedron (relative to ) of the BB-polynomial .
For a set and a BB-polynomial , denote BB-polynomial by . It is called the -truncation of .
A BB-polynomial is called nondegenerate if and any truncation on a proper face of has a nonsingular zero set in .
Let be a subset of the set of domain points in simplex and . Define the moment map associated with relative to the simplex , , by where is the complement in of the union of all its proper faces.
Definition 2. Let be a BB-polynomial with a Newton polyhedron . Then the closure of in is called chart of , where is the moment map associated with relative to the simplex .
Let be polyhedra with and for . Assume that is a continuous, piecewise linear, nonnegative convex function satisfying the following conditions:(1) all the restrictions are linear;(2) if the restriction of to an open set is linear, then this set is contained in one of the ;(3), where is the set of the domain points.Then, this function with this property is said to convexify .
Let be BB-polynomials over in variables with . Let for any . Then, there exists a unique BB-polynomial with and for . If and is a function convexifying , we put The BB-polynomials are said to be obtained by BB-polynomials by or, briefly, is a of BB-polynomials by .
Let be BB-polynomials in variables with , and let for . A chart of a BB-polynomial with is said to be obtained by charts of BB-polynomials and it is a of charts of BB-polynomials if and the chart of , up to isotopy, is .
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 ).
Proposition 3 (see ). Let , and be as above ( is a of BB-polynomials by ). If BB-polynomials are nondegenerate, then there exists such that for any the chart of BB-polynomial is obtained by patchworking charts of BB-polynomials .
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 be a BB-polynomial with Newton polyhedron and let be a face of . Then,(1);(2) if is nondegenerate with respect to (which is the case when, for example, is nondegenerate with respect to ), then, the intersects transversally.
Proof. Suppose that the BB-polynomial is defined in (4) and
Define the mapping by According to [18, Lemma 3.2], then, is the Newton polyhedron of the polynomial and is a face of if is a face of . Moreover, by [10, Remark 1.1], the of (see [6–8, 18]) and the truncation on the face have the following properties:(a);(b) the intersects transversally if is nondegenerate with respect to .
On the other hand, by [18, Lemma 3.2 and Theorem 3.5], we can get the following equalities: This, together with equality (10) and properties (a) and (b), shows that and the intersects transversally if is nondegenerate with respect to . This completes the proof.
Proposition 5 (see [3, 15]). Let be BB-polynomials of degree that are defined on two adjacent -dimensional simplices and , respectively. Then and are smoothly connected on if and only if for all and with , where , , and . Here, denotes the barycentric coordinate of with respect to the simplex .
Theorem 6. Suppose that and defined as above are smoothly connected on . Then,(1), and is a face of and if is nonempty, where and are Newton polyhedra of BB-polynomials and , respectively;(2) for above.
Proof. By assumption and taking in equality (13), we have that for all with . Thus, , where is defined in (6). This implies that , and so is a face of and if is nonempty.
According to the argument above, it is easy to see that -truncations , of and satisfy the equality . Thus, we can get that by the definition of the .
On the other hand, it follows from Theorem 4 that Therefore, by equality (14). This completes the proof.
Let be a pure, hereditary simplicial complex with -cells (i.e., -dimensional simplex) . Let and each polynomial be expressed in the B-form The piecewise polynomial defined as above is called piecewise BB-polynomial.
Theorem 7. Let be a piecewise BB-polynomial defined in (17). Then, for each adjacent pair , of -cells in ,(1), and is a face of and if is nonempty, where and are Newton polyhedra of BB-polynomials and , respectively;(2) for above.
Proof. We can get the conclusion from Theorem 6 immediately.
Definition 8. Let be a piecewise BB-polynomial defined in (17), and let be the Newton polyhedron of the BB-polynomial . The chart of is the closure of , where and is the moment map associated with relative to the simplex .
Theorem 9. Let be a piecewise BB-polynomial defined in (17). Then, the of is .
4. The Construction of Real 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  and the smooth blending theory.
Let be a pure, hereditary simplicial complex with -cells (i.e., -dimensional simplex) .
For each adjacent pair , of -cells in , assume that ,, where the hat means that the corresponding vertex is omitted. Obviously, there exist two arrangements and of numbers such that and for .
Now, we define two one-to-one transformations , , by , and , respectively, where .
For any given , let be BB-polynomials over the simplex with , , and let be a of BB-polynomials by .
Define the piecewise BB-polynomial on the domain by
Theorem 10. Suppose that are non-degenerate, that is a of BB-polynomials by , , and that , , , , , , , , and are defined as above, and for . If for each adjacent pair , of -cells in and all and with , where and denotes the barycentric coordinate of with respect to the simplex , then,(1);(2) if for any BB-polynomial with , , there exists a BB-polynomial in such that , then there is such that for any , the chart of piecewise BB-polynomial , up to isotopy, is .
Proof. Let and be the barycentric coordinates of point with respect to and , respectively. Then,
by the definition of the transformation , and so
Set . Then, and . Therefore, it follows from equality (22) that
This shows that the polynomial over can be represented as follows:
By a similar argument above, we have that the polynomial over can be represented as follows where is the barycentric coordinate of point with respect to .
Since for , ; thus, according to Proposition 5, if and only if the BB-coefficients of and the BB-coefficients of satisfy equality (20). This proves property .
Below we will show that property holds.
For each adjacent pair , of -cells in , set when ; then, is a face of and by the assumption that . Since (resp., ) is a patchworking of BB-polynomials (resp., ) by (resp., ) and for , then, for -truncations , , and of , , , and , respectively, we have that
Since and are smoothly connected on , and (20) implies that . Thus, by (26).
Using a similar argument in the proof of Theorem 6, we have by . This shows that
On the other hand, since is a patchworking of BB-polynomials by , we get from the assumption and Proposition 3 that there is such that for any the chart of BB-polynomial (i.e., ), up to isotopy, is . This conclusion together with Theorems 7–9 and equality (27) implies that there is such that for any the chart of piecewise BB-polynomial , up to isotopy, is , where . This completes the proof.
Assume that is a -dimensional simplex with domain points in as its vertices and that the BB-polynomial is a real -nomial (this means that the nonzero coefficients in correspond to the only vertices of ). Denote by the set of middle points of edges of . For any point on an edge with endpoints , , we assign that where are the corresponding BB-coefficients in . Put
Theorem 11. Suppose that ,, and satisfy the conditions in Theorem 10 and that for each , are real -nomial and with dimension form a triangulation of . Then, and there exists a constant such that for any the chart of , up to isotopy, is , where is defined as (29).
Proof. By assumption and [18, Theorem 4.4], we know that there is such that for any the chart of BB-polynomial , up to isotopy, is .
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 piecewise algebraic hypersurface with a prescribed complex topology and a degree on a partition , we can just construct some Bernstein-Bézier algebraic hypersurfaces pieces with simple topology and a convex function on each -cell , where , and satisfy the conditions of Theorem 10 or Theorem 11, , and then, the BB-polynomial is obtained by a patchworking of BB-polynomials by . Thus, we can get by defining .
Conflict of Interests
The authors do not have any possible conflict of interests in this paper.
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).
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