一类直纹面的研究

本给出了过任意空间C^k(k≥3)类光滑曲线的直纹面是可展曲面的充要条件．同时得到了该空间曲线为相应直纹面的曲率线,测地线和渐近曲线的充要条件。     

复射影直纹面（II）

本文利用几何直纹面及曲线上秩－２局部自由层的一些性质，讨论射影空间中的直纹面一些特性，给出了非正则直纹面次数的下界并讨论了个维射影空间中次数接近下界的非正则直纹面的结构，如奇点集的结构，底曲线的结构及纤维束次数等，完全确定了这类曲面的几何结构。     

关于kr-空间与k-空间

本文给出了具有局部可数闭ｋ－网的ｋｒ－空间成为ｋ－空间的一个充要条件，此结果给林寿提出的问题作了一个回答。     

对偶曲线和直线轨迹

本文介绍了拟合或设计三维空间中直线轨迹的一种新方法,利用对偶数和对偶向量,我们把直纹面对应到单位对偶球面上的曲线。证明了这种曲线的Frenet公式。在引进对偶球面坐标之后,把直纹面对应到四维空间中的一条曲线,于是Bézier曲线,B样条曲线和累加弦长参数曲线成为拟合或设计直纹面的重要手段。     

交换半环上半线性空间的直和

本文研究交换半环上半线性子空间的直和问题.首先在一定条件下给出了任意n维向量组能够扩充成半线性空间的一组基的充要条件;在此基础上,给出了两个子空间的和为直和的一些充要条件.     

三维空间形式中紧致常平均曲率曲面拓扑型的曲率特征

把陈省身教授关于三维球空间中紧致常平均曲率球面拓扑型的曲率特征的研究,以及浙江大学水乃翔教授等人关于环面的相应研究,扩展到一般三维空间形式中任意紧致定向常平均曲率由面拓扑型的曲率特征的研究．证明了曲率为ｋ的三维空间形式中的紧致定向常平均曲阜曲面Ｍ为拓扑球面的充要条件是 ｋ＋ Ｈ２－ Ｋ＝０, Ｍ为拓扑环面的充要条件是 ｈ＋ Ｈ２－ Ｋ＞ ０, Ｍ的亏格为ｇ（≥２）的充要条件是ｋ＋ｈ２－Ｋ的零点个数为８（ｇ－１）,其中Ｈ和Ｋ分别为Ｍ的平均曲率和Ｇａｕｓｓ曲率．前两个结果分别推广了陈省身和水乃翔等人的结果．     

区间上k段单调连续自映射的k阶迭代根

本文得到了区间Ｉ＝［０，１］上的ｋ段单调连续自映射具有ｋ阶迭代根的充要条件     

由给定的子空间构造新的子空间

本给出并证明了若干个子空间的并以及两个子空间的基构成子空间的充要条件，从而本质地揭示了除子空间的交与和是构造新的予空间的方法外，集合的其它运算不能构造新的子空间，最后分析了子空间直和的两种不同定义的优缺点，指出了张禾瑞教材中子空间直和定义推广时应注意的一个问题。     

非负广义幂等矩阵

非负幂等矩阵在［１］中已有详细地讨论和重要地应用。［２］给出了Ａ≥０，Ａｋ＋１＝Ａ，ｋ是正整数，Ａｋ是零对称幂等矩阵的充要条件。本文给出了一般非负广义幂等矩阵的充要条件。并讨论了广义幂等矩阵的秩与其广义幂等指数的关系，推广了［２］中的部分结论。     

线性空间中代数广义逆的最简表示

本文主要在一般线性空间框架中从纯代数的角度研究代数广义逆的可加性与表示问题.首先在线性空间中利用空间代数直和分解给出I+AT~+可逆的充要条件,进而T~+=T~+(I+A~T+)~(-1),给出了T~+具有最简表示的一系列充要条件.其次讨论了在Banach空间广义逆和Hilbert空间Moore-Penrose逆扰动问题研究中的应用.本文的主要结果推广和改进了相关文献中的一些近期成果.     

On closed timelike and spacelike ruled surfaces

In this study, by using the concepts and results on spherical curves in dual Lorentzian space, we give the criterions for ruled surfaces with non‐lightlike ruling to be closed (periodic). Moreover, we obtain the necessary and sufficient conditions to guarantee that a timelike or a spacelike ruled surface is closed (periodic). Copyright © 2016 John Wiley & Sons, Ltd.     

A short proof of the generalized Bonnet theorem

In three‐dimensional Euclidean space E³, the Bonnet theorem says that a curve on a ruled surface in three‐dimensional Euclidean space, having two of the following properties, has also a third one, namely, it can be a geodesic, that it can be the striction line, and that it cuts the generators under constant angle. In this work, in n dimensional Euclidean space Eⁿ, a short proof of the theorem generalized for (k + 1) dimensional ruled surfaces by Hagen in 4 is given. Copyright © 2013 John Wiley & Sons, Ltd.     

Sechseckgewebe und Strahlensysteme

Let S be an oriented rectilinear congruence in the three-dimensional Euclidean space E³. In this paper we prove necessary and sufficient conditions, so that certain ruled surfaces of S meet its middle surface in an hexagonal web.     

Surface pencil with a common line of curvature in Minkowski 3-space

In this paper, we analyze the problem of constructing a surface pencil from a given spacelie （timelike） line of curvature. Using the Frenet frame of the given line of curvature in Minkowski 3-space, we express the surface pencil as a linear combination of this frame and derive the necessary and sufficient conditions for the coefficients to satisfy the line of curvature requirement. We illustrate this method by presenting some examples.     

Über direkt integrierbare Kurven und strenge Böschungskurven im ℝn sowie geschlossene verallgemeinerte Hyperkreise

A curve in Euclidean space ?ⁿ is called “directly integrable”, if it can be explicitly calculated from the curvatures in a specified way. A necessary and sufficient condition for a curve to be directly integrable is that all its curvatures are real multiples of a single real function. Directly integrable curves in an odd-dimensional space ?ⁿ (n=2q+1) can be interpreted as generalized helices. In the case of even-dimensional space ?ⁿ (n=2p), we give a simple necessary and sufficient condition for a directly integrable curve to be closed.     

Intersection of a ruled surface with a free-form surface

This paper presents a simple method for computing the intersection curve of a ruled surface and a free-form surface. The basic idea is to reduce the problem of surface intersection to the one of projecting an appropriate curve such as a directrix of the ruled surface, along its indicatrix curve (direction vector field of its generating lines), onto the free-form surface; the projection curve is just the intersection curve. With techniques in classical differential geometry, we derive the differential equations of the intersection curve in the cases of parametrically and implicitly defined free-form surfaces. The intersection curve naturally inherits the parameter of the chosen directrix. Moreover, it is independent of the base surface geometry and its parameterization, and is obtained by numerically solving the initial-value problem for a system of first-order ordinary differential equations in the parametric domain associated to the surface representation for parametric case or in 3D space for implicit case. Some experimental examples are also given to demonstrate that the presented method is effective and potentially useful in computer aided design and computer graphics. An erratum to this article can be found at     

On Cremona transformations and quadratic complexes

We study the relation between Cremona transformations in space and quadratic line complexes. We show that it is possible to associate a space Cremona transformation to each smooth quadratic line complex once we choose two distinct lines contained in the complex. Such Cremona transformations are cubo-cubic and we classify them in terms of the relative position of the lines chosen. It turns out that the base locus of such a transformation contains a smooth genus two quintic curve. Conversely, we show that given a smooth quintic curve C of genus 2 in ℙ³ every Cremona transformation containing C in its base locus factorizes through a smooth quadratic line complex as before. We consider also some cases where the curve C is singular, and we give examples both when the quadratic line complex is smooth and singular.     

Approximation by ruled surfaces

Given a surface or scattered data points from a surface in 3-space, we show how to approximate the given data by a ruled surface in tensor product B-spline representation. This leads us to a general framework for approximation in line space by local mappings from the Klein quadric to Euclidean 4-space. The presented algorithm for approximation by ruled surfaces possesses applications in architectural design, reverse engineering, wire electric discharge machining and NC milling.     

Minkowski空间中定向曲面上的第二类松弛弹性线

在Minkowski空间中,定义了定向曲面上的第二类松弛弹性线,推导了在定向曲面上的第二类松弛弹性线的Euler-Lagrange方程．进一步阐明了,这些曲线是否落在曲率线上,最后给出相关的实例．     

Evolutes of dual spherical curves for ruled surfaces

E. Study found that there is a one‐to‐one correspondence between the oriented lines in Euclidean three space and the dual points of the dual unit sphere in dual three space, and it has wide applications in Engineering. In this paper, we investigate a ruled surface as a curve on the dual unit sphere by using E. Study's theory. Then we define the notion of evolutes of dual spherical curves for ruled surfaces and establish the relationships between singularities of these subjects and geometric invariants of dual spherical curves. Finally, we give an example to illustrate our findings. Copyright © 2015 John Wiley & Sons, Ltd.