Nther-type theorem of piecewise algebraic curves on triangulation

The piecewise algebraic curve is a kind generalization of the classical algebraic curve. Nther-type theorem of piecewise algebraic curves on the cross-cut partition is very important to construct the Lagrange interpolation sets for a bivariate spline space.In this paper,using the properties of bivariate splines,the Nther-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented.     

The Nother and Riemann-Roch type theorems for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, the Nother type theorems for Cμpiecewise algebraic curves are obtained. The theory of the linear series of sets of places on the piecewise algebraic curve is also established. In this theory, singular cycles are put into the linear series, and a complete series of the piecewise algebraic curves consists of all effective ordinary cycles in an equivalence class and all effective singular cycles which are equivalent specifically to any effective ordinary cycle in the equivalence class. This theory is a generalization of that of linear series of the algebraic curve. With this theory and the fundamental theory of multivariate splines on smoothing cofactors and global conformality conditions, and the results on the general expression of multivariate splines, we get a formula on the index, the order and the dimension of a complete series of the irreducible Cμpiecewise algebraic curves and the degree, the genus and the smoothness of the curves, hence the Riemann-Roch type theorem of the Cμpiecewise algebraic curve is established.     

Estimation of the Bezout number for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function.In this paper.a coniecture on trianguation is confirmed The relation between the piecewise linear algebraiccurve and four-color conjecture is also presented.By Morgan-Scott triangulation, we will show the instabilityof Bezout number of piecewise algebraic curves. By using the combinatorial optimization method,an upper     

Numerical realization of the conditions of Max NSther＇s residual intersection theorem

The aim of this paper is to study numerical realization of the conditions of Max Nother＇s residual intersection theorem. The numerical realization relies on obtaining the inter- section of two algebraic curves by homotopy continuation method, computing the approximate places of an algebraic curve, getting the exact orders of a polynomial at the places, and determin- ing the multiplicity and character of a point of an algebraic curve. The numerical experiments show that our method is accurate, effective and robust without using multiprecision arithmetic, even if the coefficients of algebraic curves are inexact. We also conclude that the computational complexity of the numerical realization is polynomial time.     

Numerical realization of the conditions of Max Nther's residual intersection theorem

The aim of this paper is to study numerical realization of the conditions of Max Nther's residual intersection theorem. The numerical realization relies on obtaining the intersection of two algebraic curves by homotopy continuation method, computing the approximate places of an algebraic curve, getting the exact orders of a polynomial at the places, and determining the multiplicity and character of a point of an algebraic curve. The numerical experiments show that our method is accurate, effective and robust without using multiprecision arithmetic,even if the coefficients of algebraic curves are inexact. We also conclude that the computational complexity of the numerical realization is polynomial time.     

The Viro method for construction of Bernstein-Bézier algebraic hypersurface piece

This paper proposes a new method for the construction of Bernstein-Bézier algebraic hypersurface on a simplex with prescribed topology.The method is based on the combinatorial patchworking of Viro method.The topology of the Viro Bernstein-Bézier algebraic hypersurface piece is also described.     

Degree elevation from Bzier curve to C-Bzier curve with corner cutting form

The existing results of curve degree elevation mainly focus on the degree of algebraic polynomials. The paper considers the elevation of degree of the trigonometric polynomial, from a Bzier curve on the algebraic polynomial space, to a C-B′ezier curve on the algebraic and trigonometric polynomial space. The matrix of degree elevation is obtained by an operator presentation and a derivation pyramid. It possesses not a recursive presentation but a direct expression. The degree elevation process can also be represented as a corner cutting form.     

球面上的几何连续插值

In this paper, a new method for geometrically continuous interpolation in spheres is proposed. The method is entirely based on the spherical B′ezier curves defined by the generalized de Casteljau algorithm. Firstly we compute the tangent directions and curvature vectors at the endpoints of a spherical B′ezier curve. Then, based on the above results, we design a piecewise spherical B′ezier curve with G 1 and G 2 continuity. In order to get the optimal piecewise curve according to two different criteria, we also give a constructive method to determine the shape parameters of the curve. According to the method, any given spherical points can be directly interpolated in the sphere. Experimental results also demonstrate that the method performs well both in uniform speed and magnitude of covariant acceleration.     

HOMOCLINIC CYCLES OF A QUADRATIC SYSTEM DESCRIBED BY QUARTIC CURVES

This paper is devoted to discussing the topological classification of the quartic invariant algebraic curves for a quadratic system. We obtain sufficient and necessary conditions which ensure that the homoclinic cycle of the system is defined by the quartic invariant algebraic curve. Finally, the corresponding global phase diagrams are drawn.     

Morgan-Scott型剖分上的样条空间的奇异性

Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over Morgan-Scott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines （or ponits）, and characteristic number of algebraic curve. With these concepts and the relevant results, a polished necessary and sufficient conditions for the singularity of spline space S u＋1^u （△MS^u） are geometrically given for any smoothness u by recursion. Moreover, the famous Pascal＇s theorem is generalized to algebraic plane curves of degree n≥3.     

实分片代数曲线的某些研究

分片代数曲线作为二元样条函数的零点集合是经典代数曲线的推广. 利用代数的基本知识, 本文对实分片代数曲线的基本性质进行了初步讨论, 并且将实分片代数曲线与相应的二元样条分类进行讨论. 最后, 对实分片代数曲线上的孤立点进行了研究.     

Nöther-type theorem of piecewise algebraic curves on triangulation

The piecewise algebraic curve is a kind generalization of the classical algebraic curve. Nöther-type theorem of piecewise algebraic curves on the cross-cut partition is very important to construct the Lagrange interpolation sets for a bivariate spline space. In this paper, using the properties of bivariate splines, the Nöther-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented.     

N(o)ther-type theorem of piecewise algebraic curves on triangulation

The piecewise algebraic curve is a kind generalization of the classical algebraic curve.N(o)ther-type theorem of piecewise algebraic curves on the cross-cut partition is very important to construct the Lagrange interpolation sets for a bivariate spline space. In this paper, using the properties of bivariate splines, the N(o)ther-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented.     

Nöther-type theorem of piecewise algebraic curves on quasi-cross-cut partition

Nöther’s theorem of algebraic curves plays an important role in classical algebraic geometry. As the zero set of a bivariate spline, the piecewise algebraic curve is a generalization of the classical algebraic curve. Nöther-type theorem of piecewise algebraic curves is very important to construct the Lagrange interpolation sets for bivariate spline spaces. In this paper, using the characteristics of quasi-cross-cut partition, properties of bivariate splines and results in algebraic geometry, the Nöther-type theorem of piecewise algebraic curves on the quasi-cross-cut is presented.     

The Nöther and Riemann-Roch type theorems for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, the Nöther type theorems for C ^µ piecewise algebraic curves are obtained. The theory of the linear series of sets of places on the piecewise algebraic curve is also established. In this theory, singular cycles are put into the linear series, and a complete series of the piecewise algebraic curves consists of all effective ordinary cycles in an equivalence class and all effective singular cycles which are equivalent specifically to any effective ordinary cycle in the equivalence class. This theory is a generalization of that of linear series of the algebraic curve. With this theory and the fundamental theory of multivariate splines on smoothing cofactors and global conformality conditions, and the results on the general expression of multivariate splines, we get a formula on the index, the order and the dimension of a complete series of the irreducible C ^µ piecewise algebraic curves and the degree, the genus and the smoothness of the curves, hence the Riemann-Roch type theorem of the C ^µ piecewise algebraic curve is established.     

The Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at mnT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m + n − 2 containing all but one point of them also contains the last point.     

Real intersection points of piecewise algebraic curves

The piecewise algebraic curve, as the set of zeros of a bivariate spline function, is a generalization of the classical algebraic curve. In this work, we present an algorithm for computing the real intersection points of piecewise algebraic curves. It is primarily based on the interval zeros of the univariate interval polynomial in Bernstein form. An illustrative example is provided to show that the proposed algorithm is flexible.     

The correspondence between multivariate spline ideals and piecewise algebraic varieties

As a piecewise polynomial with a certain smoothness, the spline plays an important role in computational geometry. The algebraic variety is the most important subject in classical algebraic geometry. As the zero set of multivariate splines, the piecewise algebraic variety is a generalization of the algebraic variety. In this paper, the correspondence between piecewise algebraic varieties and spline ideals is discussed. Furthermore, Hilbert’s Nullstellensatz for the piecewise algebraic variety is also studied.