Piecewise Algebraic Curves Research Articles

The maximum number and its distribution of singular points for parametric piecewise algebraic curves

The piecewise algebraic curve, as the zero set of a bivariate spline function, is a generalization of the classical algebraic curve. Based on the previous method presented by Lai et al. (2009), we show that computing singular points of parametric piecewise algebraic curves amounts to solving parametric piecewise polynomial systems. In this article, we give a method to compute the maximum number and its distribution of singular points for a given parametric piecewise algebraic curve. This method also produces necessary and sufficient conditions of its parameters must be satisfied. An illustrated example shows that our method is flexible.

Differential Equations and Exact Solutions in the Moving Sofa Problem

ABSTRACTThe moving sofa problem, posed by Moser in 1966, asks for the planar shape of maximal area that can move around a right-angled corner in a hallway of unit width and is conjectured to have as its solution a complicated shape derived by Gerver in 1992. We extend Gerver's techniques by deriving a family of six differential equations arising from the area-maximization property. We then use this result to derive a new shape that we propose as a possible solution to the ambidextrous moving sofa problem, a variant of the problem previously studied by Conway and others in which the shape is required to be able to negotiate a right-angle turn both to the left and to the right. Unlike Gerver's construction, our new shape can be expressed in closed form, and its boundary is a piecewise algebraic curve. Its area is equal to , where X and Y are solutions to the cubic equations x2(x + 3) = 8 and x(4x2 + 3) = 1, respectively.

Bounds on the number of solutions of polynomial systems and the Betti numbers of real piecewise algebraic hypersurfaces

This paper, based on Bihan and Sottile’s method which reduces a polynomial system to its Gale dual system and then bounds the number of solutions of this Gale system, proves that a real coefficient polynomial system with n equations and with n variables involving n + k + 1 monomials has fewer than 27 e 5 3 + 88 90 ∏ i = 0 k − 1 ( 2 i ( n − 1 ) + 1 ) positive solutions and 27 e 10 3 + 88 90 ∏ i = 0 k − 1 ( 2 i ( n − 1 ) + 1 ) non-degenerate non-zero real solutions. This dramatically improves F. Bihan and F. Sottile’s bounds of e 2 + 3 4 2 k 2 n k and e 4 + 3 4 2 k 2 n k respectively. Using the new upper bound for positive solutions, we establish restrictions to the sum of the Betti numbers of real piecewise algebraic hypersurfaces and real piecewise algebraic curves. A new bound on the number of compact components of algebraic hypersurfaces in R > n is also given.

Construction of point-tangent interpolating algebraic curve

Algebraic curve plays an important role in geometric modelling. In order to improve algebraic curve’s flexibility, we construct point-tangent interpolating piecewise algebraic curve with shape parameters. We first construct degree three and degree four point-tangent interpolating algebraic curves when the number of the given points is three or four. The continuity of the curve is also provided. We then discuss the issue of smooth joint between curve segments. Finally, some practical examples are given to show the effectiveness of this new method.

Survivable minimum bottleneck networks

We show that the survivable bottleneck Steiner tree problem in normed planes can be solved in polynomial time when the number of Steiner points is constant. This is a fundamental problem in wireless ad-hoc network design where the objective is to design networks with efficient routing topologies. Our result holds for a general definition of survivability and for any norm whose ball is specified by a piecewise algebraic curve of bounded degree and with a bounded number of pieces. In particular, under the Euclidean and rectilinear norms our algorithm constructs an optimal solution in O(n2k+3log⁡n) steps, where n is the number of terminals and k is the number of Steiner points. Our algorithm is based on the construction of generalised Voronoi diagrams and relative neighbourhood graphs.

An upper bound of the Bezout number for piecewise algebraic curves

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. Based on the discussion of the number of the zeros of homogeneous trigonometric splines with different smoothness and the common points of two piecewise algebraic curves over a star partition, a better upper bound of Bezout number of two piecewise algebraic curves over arbitrary triangulation is found. Moreover, upper bounds of the Bezout number BN(m,r;n,r;Δ) for piecewise algebraic curves over several special partitions such as rectangular partition, type-1 triangulation and type-2 triangulation are obtained.

The Bezout number of piecewise algebraic curves

Based on the discussion of the number of roots of univariate spline and the common zero points of two piecewise algebraic curves, a lower upbound of Bezout number of two piecewise algebraic curves on any given non-obtuse-angled triangulation is found. Bezout number of two piecewise algebraic curves on two different partitions is also discussed in this paper.

An Upper Bound of the Bezout Number for Piecewise Algebraic Curves over a Rectangular Partition

A piecewise algebraic curve is a curve defined by the zero set of a bivariate spline function. Given two bivariate spline spaces (Δ) over a domainDwith a partition Δ, the Bezout number BN(m,r;n,t;Δ) is defined as the maximum finite number of the common intersection points of two arbitrary piecewise algebraic curves (Δ). In this paper, an upper bound of the Bezout number for piecewise algebraic curves over a rectangular partition is obtained.

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 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.

An Algorithm for Isolating the Real Solutions 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 paper, an algorithm is presented to compute the real solutions of two piecewise algebraic curves. It is primarily based on the Krawczyk‐Moore iterative algorithm and good initial iterative interval searching algorithm. The proposed algorithm is relatively easy to implement.

Estimate of The Bézout Number For Linear Piecewise Algebraic Curves Over Arbitrary Triangulations

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. This paper gives an upper bound of the Bézout number, the maximum number of intersections between two linear piecewise algebraic curves whose intersections are finite, over arbitrary triangulations.

The Bezout number for linear piecewise algebraic curves

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. This paper discusses the Bezout number, the maximum number of intersections between two linear piecewise algebraic curves whose intersections are finite, on regular triangulations. We give an upper bound of the Bezout number for linear piecewise algebraic curves ( B N ( 1 , 0 ; 1 , 0 ; Δ ) ) on the triangulation with an odd interior vertex. For the triangulations which satisfy a vertex coloring condition, we compute the exact value of the Bezout number B N ( 1 , 0 ; 1 , 0 ; Δ ) .

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

Nother’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. Nother-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 Nother-type theorem of piecewise algebraic curves on the quasi-cross-cut is presented.

Lagrange interpolation set along linear piecewise algebraic curves

This paper discusses the Lagrange interpolation problem in continuous bivariate spline spaces over regular triangulations. By using the so-called Lagrange interpolation set along piecewise algebraic curves, we develop a new approach of constructing the interpolation set for continuous spline spaces. We show the property of this set on star region, and construct the interpo- lation set for continuous bivariate spline spaces over arbitrary triangulations. The construction only depends on the number of points on the piecewise algebraic curve in each cell.

Approximate Implicitization of Parametric Curves Using Cubic Algebraic Splines

This paper presents an algorithm to solve the approximate implicitization of planar parametric curves using cubic algebraic splines. It applies piecewise cubic algebraic curves to give a global G2 continuity approximation to planar parametric curves. Approximation error on approximate implicitization of rational curves is given. Several examples are provided to prove that the proposed method is flexible and efficient.

Intersection points algorithm for piecewise algebraic curves based on Groebner bases

Piecewise algebraic curve is defined as the zero set of a bivariate spline. In this paper, we mainly study the intersection points algorithm for two given piecewise algebraic curves based on Groebner bases. Given a domain D and a partition Δ, we present a flow and introduce the truncated signs, and then represent the two piecewise algebraic curves in the global form. We get their Groebner bases with respect to a lexicographic order and adopt the interval arithmetic in the back-substitution process, which makes the algorithm numerically precise. An example is also presented to show the algorithm’s feasibility and effectiveness.

Counting positive solutions for polynomial systems with real coefficients

Based on the subresultant polynomial chain and the method presented by Hongguang Fu and Lu Yang et al., we propose a practical algorithm for computing the number of sign changes of the leading coefficients and the constant terms of a generalized Sturm sequence (GSS). With the principal minors sequence of the discrimination matrix of the polynomial and the number of sign changes of the GSS, a formula determining the number of positive solutions of a given bivariate polynomial systems with real coefficients (with a finite number of complex solutions) and its algorithm are presented. Using the techniques of the B - net form of bivariate splines function, this algorithm can be used for computing the number of real intersection points of two piecewise algebraic curves and its distributions in all cells when common points are finite. A piecewise algebraic curve is defined by a bivariate spline function.

Nöther-type theorem of piecewise algebraic curves on triangulation

The piecewise algebraic curve is a kind generalization of the classical algebraic curve. Nother-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 Nother-type theorem of piecewise algebraic curves on the arbitrary triangulation 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 Nother type theorems for Cµ 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 Cµ piecewise algebraic curves and the degree, the genus and the smoothness of the curves, hence the Riemann-Roch type theorem of the Cµ piecewise algebraic curve is established.