Modifying the shape of FB-spline curves

FB-spline curves are the unification of recently developed trigonometric CB-spline and hyperbolic HB-spline curves, including the classical B-spline curves. These generalized curves overcome some restrictions of B-spline curves and allow to design some important curves like helix, cycloids or catenary. Their properties, however, have been studied only theoretically. In this paper practical shape modification algorithms of FB-spline curves are discussed, including the geometrical effects of the alteration of shape parameters, which are essential from the users’ point of view.     

Metric Sample Spaces of Continuous Geometric Curves and Estimation of Their Centroids

The metric sample space of Fréechet curves (Fréechet , 1934,1951,1961) is based on a generalization of regular curves that covers continuous curves in full generality. This makes it possible to deal with both smooth and non-smooth, even non-rectifiable geometric curves in statistical analysis. In the present paper this sample space is further extended in two directions that are relevant in practice: to incorporate information on landmark points in the curves and to impose invariance with respect to an arbitrary group of isometric spatial transformations. Properties of the introduced sample spaces of curves are studied, specially those concerning to the generation and representation of random curves by random functions.In order to provide measures of central tendency and dispersion of random curves, centroids and restricted centroids ofrandom curves are defined in a general metric framework, and methods for their consistent estimation are derived.     

Square-tiled surfaces and rigid curves on moduli spaces

We study the algebro-geometric aspects of Teichmüller curves parameterizing square-tiled surfaces with two applications.(a) There exist infinitely many rigid curves on the moduli space of hyperelliptic curves. They span the same extremal ray of the cone of moving curves. Their union is a Zariski dense subset. Hence they yield infinitely many rigid curves with the same properties on the moduli space of stable n-pointed rational curves for even n.(b) The limit of slopes of Teichmüller curves and the sum of Lyapunov exponents for the Teichmüller geodesic flow determine each other, which yields information about the cone of effective divisors on the moduli space of curves.     

两类新的广义Ball曲线曲面的求值算法及其应用

本文研究两类新的广义Ball曲线曲面的求值算法及其应用.其一是把Bezier曲线曲面的求值转换到这两类曲线曲面的求值,大大加快了计算速度.其二是给出Bezier曲线与这两类广义Ball曲线的统一表示,并利用这种表示给出它们之间相互转换的递归算法.     

On stability crossing curves for general systems with two delays

For the general linear scalar time-delay systems of arbitrary order with two delays, this article provides a detailed study on the stability crossing curves consisting of all the delays such that the characteristic quasipolynomial has at least one imaginary zero. The crossing set, consisting of all the frequencies corresponding to all the points in the stability crossing curves, are expressed in terms of simple inequality constraints and can be easily identified from the gain response curves of the coefficient transfer functions of the delay terms. This crossing set forms a finite number of intervals of finite length. The corresponding stability crossing curves form a series of smooth curves except at the points corresponding to multiple zeros and a number of other degenerate cases. These curves may be closed curves, open ended curves, and spiral-like curves oriented horizontally, vertically, or diagonally. The category of curves are determined by which constraints are violated at the two ends of the corresponding intervals of the crossing set. The directions in which the zeros cross the imaginary axis are explicitly expressed. An algorithm may be devised to calculate the maximum delay deviation without changing the number of right half plane zeros of the characteristic quasipolynomial (and preservation of stability as a special case).     

Dynamic PDE parametric curves

Dynamic partial differential equation (PDE) parametric curves which can be expressed as a coupled system of two hyperbolic equations are developed. In curve design, dynamic PDE parametric curves can be modified intuitively and are more flexible than ordinary differential equation (ODE) curves. The calculation of dynamic PDE parametric curves must recur to numerical methods and a three-level finite difference scheme is proposed. Approximation and stability properties for the scheme are proved and convergence property is derived. An example of interpolating PDE curves is presented as an application of dynamic PDE parametric curves.     

A birational embedding with two Galois points for quotient curves

A criterion for the existence of a birational embedding with two Galois points for quotient curves is presented. We apply our criterion to several curves, for example, some cyclic subcovers of the Giulietti–Korchmáros curve or of the curves constructed by Skabelund. New examples of plane curves with two Galois points are described, as plane models of such quotient curves.     

C2 Continuous Quartic Hermite Spline Curves with Shape Parameters

In order to relieve the deficiency of the usual cubic Hermite spline curves, the quartic Hermite spline curves with shape parameters is further studied in this work. The interpolation error and estimator of the quartic Hermite spline curves are given. And the characteristics of the quartic Hermite spline curves are discussed. The quartic Hermite spline curves not only have the same interpolation and conti-nuity properties of the usual cubic Hermite spline curves, but also can achieve local or global shape adjustment and C2 continuity by the shape parameters when the interpolation conditions are fixed.     

One-side peano curves of fractal genus 9

This paper completes the analysis (begun by E.V. Shchepin and the author in 2008) of regular Peano curves of genus 9 in search of a curve with the minimum square-to-linear ratio. One-side regular Peano curves of genus 9 are considered, and, among these curves, a class of minimal curves with a square-to-linear ratio of 5 2/3 is singled out. A new language to describe curves is introduced which significantly simplifies the coding of these curves.     

基于面积约束的统计直方图曲线拟合

我们提出用分段三次Hermite插值曲线拟合统计直方图的新方法.先根据统计直方图的特点选取Hermite插值曲线在插值点处的导数值和可调整的插值点,然后根据面积约束确定调整值,从而得到拟合曲线.所得拟合曲线与统计直方图有面积相等的约束,并且拟合曲线是C1连续的光滑曲线.所给方法简单、实用.     

二阶曲线间的射影映射

平面上的射影变换,将二阶曲线变为另一二阶曲线,这个射影变换也可以称为这两个二阶曲线间的射影映射.若两个二阶曲线相切,则存在以切点为射影中心的两个二阶曲线间的射影映射;若两个二阶曲线相离,则存在以两个二阶曲线公切线交点为射影中心的射影映射;若两个二阶曲线相交,则存在以其中一交点为射影中心的两个二阶曲线间的射影映射.     

Maximal and Cohen–Macaulay projective monomial curves

In this paper we continue the investigation of Cohen–Macaulay projective monomial curves begun in [Les Reid, Leslie G. Roberts, Non-Cohen–Macaulay projective monomial curves, J. Algebra 291 (2005) 171–186]. In the process we introduce maximal curves. Cohen–Macaulay curves are maximal, but not conversely. We show that the number of all curves of degree d that are Cohen–Macaulay grows exponentially, but not as fast as the total number of curves, and also that maximal curves of degree d with sufficiently large embedding dimension relative to d are Cohen–Macaulay.     

Curves on normal surfaces

We study locally Cohen-Macaulay space curves lying on normal surfaces. We prove some theorems on the behaviour of the cohomology functions and initial ideals of such space curves, which give a basic distinction between those curves and curves lying on non-normal surfaces.

     

与给定的多边形相切的闭(C~2-连续)Bézier曲线

Bezier曲线已广泛应用到汽车、航空、造船等许多工业领域中.[1]描述了以给定凸多边形的所有边为切线的.平面三次分段C~2-连续的闭Bezier曲线的构造,并且给出了实际应用. [1]描述的算法有如下缺点:(a)需要解一个大型线性方程组,计算量很大;(b)对     

A duality theorem for crossed products of hopf algebras

Abstract

We study the automorphism and collineation groups of plane curves, i.e., in ?², that are not necessarily smooth, and obtain bounds for these curves in terms of, their degree and the number of singularities. We also introduce the notion of bad curves and good curves, and show that all bad curves are rational.     

Invariant curves for a second-order difference equation modelled from macroeconomics

In this paper we investigate invariant curves of a planar mapping, which equivalently formulates a second-order difference equation modelled from macroeconomics. We construct all invariant curves in the linear case and prove the existence of invariant curves in the nonlinear case. Furthermore, we discuss the continuous dependence of invariant curves in the nonlinear case.     

Seshadri constants on Jacobian of curves

We compute the Seshadri constants on the Jacobian of hyperelliptic curves, as well as of curves with genus three and four. For higher genus curves we conclude that if the Seshadri constants of their Jacobian are less than 2, then the curves must be hyperelliptic.

     

The Dual Functionals for the Generalized Ball Basis of Wang-Said Type and Basis Transformation Formulas

The generalized Ball curves of Wang-Said type with a position parameter L not only unify the Wang-Ball curves and the Said-Ball curves, but also include several useful intermediate curves. This paper presents the dual functionals for the generalized Ball basis of Wang-Said type. The relevant basis transformation formulae are also worked out.     

Equiframed Curves – A Generalization of Radon Curves

Equiframed curves are centrally symmetric convex closed planar curves that are touched at each of their points by some circumscribed parallelogram of smallest area. These curves and their higher-dimensional analogues were introduced by Peczynski and Szarek (1991, Math Proc Cambridge Philos Soc 109: 125–148). Radon curves form a proper subclass of this class of curves. Our main result is a construction of an arbitrary equiframed curve by appropriately modifying a Radon curve. We give characterizations of each type of curve to highlight the subtle difference between equiframed and Radon curves and show that, in some sense, equiframed curves behave dually to Radon curves.