On evolutoids of planar convex curves

There are many interesting curves which we can associate with a given convex curve, however, in this work we are especially interested in studying the relations between the given curve and its evolutoids: that is, the curve obtained as the envelope of lines making a fixed angle with the normal line at every point of the curve. The first result is an inequality between the area enclosed by the given curve and the area enclosed by its evolutoid. Also, we proved that a convex curve is of constant width (centrally symmetric) if and only if its evolutoid for a fixed angle is of constant width (centrally symmetric).     

On the Hyperbolicity Locus of a Real Curve

Given a real algebraic curve in the projective 3-space, its hyperbolicity locus is the set of lines with respect to which the curve is hyperbolic. We give an example of a smooth irreducible curve whose hyperbolicity locus is disconnected but the connected components are not distinguished by the linking numbers with the connected components of the curve.     

A near-optimal manhattan channel router for a class of nets with the shift-right-one pattern

The Manhattan channel routing problem of providing a one-to-one rectilinear wire connection (horizontal segments on one layer, while vertical segments are on the other) between two sets of terminals which lie on two horizontal lines, with the smallest width, is NP-hard. In this paper, it is studied for m consecutive shift-right-one nets, and a near-optimal router which produces a layout with width at most 2 more than the theoretical lower bound given in [1] is presented.     

Galois lines for the Artin–Schreier–Mumford curve

The arrangement of all Galois lines for the Artin–Schreier–Mumford curve in the projective 3-space is described. Surprisingly, there exist infinitely many Galois lines intersecting this curve.     

Real algebraic curves of constant width

Rabinowitz constructed a parametric curve of constant width and expressed it as a plane algebraic curve; however, the algebraic curve also contains isolated points separate from the original curve. We show how to modify his example in order to produce a curve with no isolated points. We then conjecture a method for producing a family of such curves and prove the conjecture in several cases.     

管状曲面在曲线论中的应用

研究了围绕曲线的管状曲面上的曲率线,渐近线与测地线,给出它们的方程,揭示了这些曲线与Bertrand曲线或Mannheim曲线之间的关系,采用新的方法给出一条曲线是Bertrand曲线或Mannheim曲线的充要条件的另一种证明以及Mannheim侣线的曲率与挠率之间的关系．     

On the geometry of the planar 4-bar mechanism

The paper seeks to elucidate the geometry of a simple engineering mechanism, comprising four bars smoothly jointed together to form a movable quadrilateral with one fixed side. The configurations of this mechanism correspond to the points of an elliptic curve, to which is associated interesting geometry and Morse theory. By appropriate projection, this curve yields the 2-parameter family of plane curves described by points rigidly attached to the side of the quadrilateral opposite the fixed side: the geometry of the general projection is related to the configuration of lines on a Segre quartic surface.     

正螺面的两个特征

通过对曲线主法线面的研究,得到了正螺面的两个特征.     

有关法曲率的若干性质

以[2]中经典微分几何问题为切入点,运用复数与三角工具广泛深入地探讨了“过曲面上一点有n条切线,若相邻两条切线的交角为2nπ,曲面法线与切线所定平面截得曲线的曲率半径为ρ1,ρ2,ρ3,…,ρn时,∑ni=11ρim,∑ni=1ρi,∏ni=1ρi的结果”,得到了法曲率与相关的三个有趣定理.     

限制在两平行线间的等周问题

本文研究了以两平行线为支撑线的等周曲线问题.利用变分的方法,求出了以平面中两条平行线为支撑线且所围面积为定值的闭凸曲线中具有最短周长的曲线.     

On solvability of the jump problem

Let Γ be a closed non-rectifiable Jordan curve on the complex plane C. We consider the so-called jump problem, i.e. the boundary value problem for determination of a holomorphic in function with a given jump on Γ. The main result is a condition of solvability of the problem in terms of a new metric dimension of the curve.     

Uniform bounds on the number of rational points of a family of curves of genus 2

We exhibit a genus-2 curve defined over which admits two independent morphisms to a rank-1 elliptic curve defined over . We describe completely the set of -rational points of the curve and obtain a uniform bound on the number of -rational points of a rational specialization of the curve for a certain (possibly infinite) set of values . Furthermore, for this set of values we describe completely the set of -rational points of the curve . Finally, we show how these results can be strengthened assuming a height conjecture of Lang.     

一类参数曲线正则性的判别

Bézier曲线的正则性,完全由它的控制顶点决定.理想的情况是由Bézier曲线的控制顶点的几何关系,就可以判断它的正则性.本文由Bézier曲线的导矢曲线在[0,1]不等于零这些代数条件,推导出了与之等价的Bézier曲线的控制顶点之间的几何关系,即只需知道顶点之间的相对位置或计算相邻线段的斜率就可快速判断Bézier曲线的正则性.最后给出了数值例子.     

求代数曲线斜渐近线的一种新方法

根据对渐近线的定义的理解,给出了求代数曲线斜渐近线的一种新方法．     

Boundedness of bilinear multipliers whose symbols have a narrow support

This work is devoted to studying the boundedness on Lebesgue spaces of bilinear multipliers on ? whose symbol is narrowly supported around a curve (in the frequency plane). We are looking for the optimal decay rate (depending on the width of this support) for exponents satisfying a sub-Hölder scaling. As expected, the geometry of the curve plays an important role, which is described. This has applications to the bilinear Bochner-Riesz problem (in particular, boundedness of multipliers whose symbol is the characteristic function of a set), as well as to the bilinear restriction-extension problem.     

Arcs and Curves over a Finite Field

In [11], a new bound for the number of points on an algebraic curve over a finite field of odd order was obtained, and applied to improve previous bounds on the size of a complete arc not contained in a conic. Here, a similar approach is used to show that a complete arc in a plane of even order q has size q+2 or or less than . To obtain this result, first a new characterization of a Hermitian curve for any square q is given; more precisely, it is shown that a curve of sufficiently low degree has a certain upper bound for the number of its rational points with equality occurring in this bound only when the curve is Hermitian. Finally, another application is given concerning the degree of the curve on which a unital can lie.     

Curves of medium genus with many points

The defect of a curve over a finite field is the difference between the number of rational points on the curve and the Weil–Serre upper bound for the number of points on the curve. We present algorithms for constructing curves of genus 5, 6, and 7 with small defect. Our aim is to be able to produce, in a reasonable amount of time, curves that can be used to populate the online table of curves with many points found at manypoints.org.     

平面常宽凸体的构造

该文首先定义了一类平面曲线"杠杆轮"与它的臂函数,并利用臂函数给出杠杆轮的参数表示.其次,证明了杠杆轮是平面常宽曲线的一种等价刻画.最后,表明Reuleaux多边形是臂函数为分段常函数的一类杠杆轮,进而构造出偶数边的Reuleaux多边形.     

Galois lines for the Giulietti–Korchmáros curve

We describe the arrangement of all Galois lines for the Giulietti–Korchmáros curve in the projective 3-space. As an application, we determine the set of all Galois points for a plane model of the GK curve. This curve possesses many Galois points.