空间曲线在一般平面上投影方程的求法

介绍了两种求空间曲线在一般平面上投影曲线方程的方法,方法一是将投影曲线看作是柱面与一般平面的交线,而方法二是将投影曲线看作是空间曲线上各点在一般平面上的投影点组成的曲线.     

递归曲线的矩阵表示和构造方法

本文给出了递归曲线的矩阵表示和构造W曲线以及L曲线的比例因子方法．揭示了Bernstein基函数和等距B样条函数以及不等距重节点B样条函数之间的一种简单的内在关系．     

曲线积分的换元法

给出了曲线积分的换元法,丰富了曲线积分的计算方法.     

一类非端点插值B样条曲线降阶的方法

降阶算法是B样条曲线和曲面设计的一个基本算法,它广泛应用于组合曲线,蒙皮或扫描曲面等设计中.Piegl与Tiller曾给出B样条曲线的降阶方法.本文给出了解决更一般的非端点插值B样条曲线降阶的方法.新的方法主要是通过对现有的节点插入方法进行分析,给出了一种端点插值递推公式,并利用此公式对Piegl与Tiller降阶方法加以改进,使之能够解决非端点插值均匀及非均匀B样条曲线的降阶问题.     

有理曲线的多项式逼近

利用曲线摄动的思想给出了用多项式曲线逼近有理曲线的一种新方法．其基本步骤是对有理曲线的控制顶点进行摄动，使之产生一多项式曲线，并使摄动误差在某种范数意义之下达到最小．同时，通过适当控制摄动曲线的顶点，使逼近多项式曲线与有理曲线在两端点保持一定的连续性．这一结果可以与细分（ｓｕｂｄｉｖｉｓｉｏｎ）技术结合给出有理曲线的整体光滑的分片多项式逼近．实例表明，在某些情况下本文中的方法要优于传统的Ｈｅｒｍｉｔｅ插值方法及Ｔ．Ｗ．Ｓｅｄｅｒｂｅｒｇ和Ｍ．Ｋａｋｉｍｏｔｏ（１９９１）提出的杂交曲线逼近算法．     

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．     

空间曲线在坐标面上投影方程定义的研析

本文探讨了空间曲线在坐标面上的投影曲线方程的定义,对同济七版《高等数学》教材给出的空间曲线在坐标面上的投影曲线方程的定义中的"包含"关系给出了明确的解释.     

C-Bézier和B-型样条闭曲线光滑性研究

本文研究了与多边形相切的样条曲线的构造方法和基本属性问题,给出了曲线光顺度的一般定义和计算方法.利用该方法对分段C-Bézier曲线、4-5-5-4次交错B-样条曲线和3阶B样条曲线的光顺度进行计算,获得了3阶B样条曲线最为光顺的结果.     

STUDY OF THE SMOOTH DEGREE OF THE CLOSED C-B（E）ZIER AND B-TYPE SPLINE CURVE

本文研究了与多边形相切的样条曲线的构造方法和基本属性问题,给出了曲线光顺度的一般定义和计算方法.利用该方法对分段C-Bézier曲线、4-5-5-4次交错B-样条曲线和3阶B样条曲线的光顺度进行计算,获得了3阶B样条曲线最为光顺的结果.     

T-B样条曲线及其应用

给出一种基于三角函数的类B样条设计方法,称其为 T B样条,它具有 B样条曲线曲面的主要优点,它还能够无需有理形式即可精确表示圆弧、椭圆弧等二次曲线弧以及球面、椭球面等二次曲面片.     

Enriched spin curves on stable curves with two components

In [7], Mainò constructed a moduli space for enriched stable curves, by blowing-up the moduli space of Deligne–Mumford stable curves. We introduce enriched spin curves, showing that a parameter space for these objects is obtained by blowing-up the moduli space of spin curves. The author was partially supported by CNPq (Proc.151610/2005-3) and by Faperj (Proc.E-26/152-629/2005).     

Curves with finite turn

In this paper we study the notions of finite turn of a curve and finite turn of tangents of a curve. We generalize the theory (previously developed by Alexandrov, Pogorelov, and Reshetnyak) of angular turn in Euclidean spaces to curves with values in arbitrary Banach spaces. In particular, we manage to prove the equality of angular turn and angular turn of tangents in Hilbert spaces. One of the implications was only proved in the finite dimensional context previously, and equivalence of finiteness of turn with finiteness of turn of tangents in arbitrary Banach spaces. We also develop an auxiliary theory of one-sidedly smooth curves with values in Banach spaces. We use analytic language and methods to provide analogues of angular theorems. In some cases our approach yields stronger results (for example Corollary 5.12 concerning the permanent properties of curves with finite turn) than those that were proved previously with geometric methods in Euclidean spaces. The author was partially supported by the grant GAČR 201/03/0931 and by the NSF grant DMS-0244515.     

Maximality of Ciani curves over finite fields

In this paper, we will study Ciani curves in characteristic $p\ge 3$, in particular their standard forms $C:x^4+y^4+z^4+r{x}^2{y}^2+s{y}^2{z}^2+t{z}^2{x}^2=0$. It is well-known that any Ciani curve is a non-hyperelliptic curve of genus 3, and its Jacobian variety is isogenous to the product of three elliptic curves. As a main result, we will show that if C is superspecial, then $r,s,t$ belong to ${\mathbb{F}}_{p^2}$ and C is maximal or minimal over ${\mathbb{F}}_{p^2}$. Moreover, in this case we will provide a simple criterion in terms of $r,s,t,p$ that tells whether C is maximal (resp. minimal) over ${\mathbb{F}}_{p^2}$.     

一类Lorentz曲线的本质属性与拟合研究

给出曲线族L(p)=p~α(1-(1-p)~β)~γ,0β1是Lorentz曲线的充分必要条件为:α≥0,γ≥0且α+γ≥1.结果是已有相关结论的推广.在此基础上,在一类Lorentz曲线族中选择了一条最佳曲线作为衡量国家收入分配的指标.用2013全国研究生数学建模竞赛E题数据,得到了相当精确的结果(均方误差精确到10~(-6).     

The canonical model of a singular curve

We give refined statements and modern proofs of Rosenlicht’s results about the canonical model C′ of an arbitrary complete integral curve C. Notably, we prove that C and C′ are birationally equivalent if and only if C is nonhyperelliptic, and that, if C is nonhyperelliptic, then C′ is equal to the blowup of C with respect to the canonical sheaf ω. We also prove some new results: we determine just when C′ is rational normal, arithmetically normal, projectively normal, and linearly normal.      

Curves covered by the Hermitian curve

A family of maximal curves is investigated that are all quotients of the Hermitian curve. These curves provide examples of curves with the same genus, the same automorphism group and the same Weierstrass semigroup at a generic point, but that are not isomorphic.     

六次PH曲线C~1 Hermite插值

本文讨论六次PH(pythagorean hodograph)曲线的Hermite插值问题.六次PH曲线可以分为两种类型,本文使用参数曲线的复数表示形式,分别给出这两类曲线的构造方法.在给定C1连续的Hermite条件下,需要指定一个自由参数以确定插值曲线,本文进一步阐述这个自由参数的几何意义.由于六次PH曲线是非正则曲线,对于第一类曲线,不易控制奇异点在曲线中的位置;而对于第二类曲线,奇异点可以在构造过程中显式地被指定,因此可以有效地避免其在特定曲线段上的出现.     

Self-contracted curves in Riemannian manifolds

It is established that every self-contracted curve in a Riemannian manifold has finite length, provided its image is contained in a compact set.     

Le relazioni irreduttibili fra gľinvarianti d’una forma qualunque Dľottavo ordine

Let C ⊂ P ³ be a smooth and connected projective curve of genus g ≥ 2. Here in characteristic ≠ 2 we prove that its generic plane section is in uniform position and that its numerical character is connected. Partially supported by MIUR and GNSAGA of INdAM (Italy)