Circular quartics in the isotropic plane generated by projectively linked pencils of conics

A curve in the isotropic plane is circular if it passes through the absolute point F. Its degree of circularity is defined as the number of its intersection points with the absolute line f falling into the absolute point F. A curve of order four can be obtained as a locus of the intersections of corresponding conics of projectively linked pencils of conics. In this paper the conditions that the pencils and the projectivity have to fulfill in order to obtain a circular quartic of a certain degree of circularity have been determined analytically. The quartics of all degrees of circularity and all types (depending on their position with respect to the absolute figure) can be constructed using these results. The results have first been stated for any projective plane and then their isotropic interpretation has been given.     

Immobilization of smooth convex figures

We establish a curvature criterion to decide whether three points immobilize a plane convex figure with smooth boundary. Then we use it to prove in the affirmative the convex case of Kuperberg's Conjecture. Namely, we prove that any convex figure with smooth boundary, different from a circular disk, can be immobilized with three points.     

On a finite crack partially penetrating two circular inhomogeneities and some related problems

We investigate a Mode-III finite slit crack partially penetrating two circular inhomogeneities embedded in an unbounded matrix. In order to obtain analytical solutions, it is assumed that the two circular inhomogeneity-matrix interfaces are Apollonius circles with respect to the two crack tips (or equivalently the two crack tips are just mutually image points with respect to each one of the two circular interfaces). Particularly closed-form expressions of the stress intensity factors at the two crack tips are obtained even though only series form solutions to the original boundary value problem can be derived. The loadings considered in this research include: (i) remote uniform anti-plane shearing; (ii) a straight screw dislocation at any position of the three-phase composite system; (iii) a Zener-Stroh crack. The results are verified by comparison with existing solutions. The related problem of a circular hole partially merged in two circular inhomogeneities is also addressed, with closed-form expressions of the stress concentration factors derived.     

A generalization of the circumcircle theorem

The theorem relating the bisectors of the edges of a triangle and the corresponding circumscribing circle is established as a special case of a theorem for triangles with weighted vertices where the edges are partitioned with circular arcs in the proportions of the weights. The circular arcs are established as being uniquely determined by the weights and the triangle, and are given by three circles with collinear centres. These circles either intersect in zero, one or two real points, these latter points being the triple points.     

受弹性点支的任意形状的膜的振动

本文提供了一个求解受弹性点支的任意形状的膜的振动的新方法.将弹性点支反力看作是作用于膜上的未知外力,求出了包含有未知反力的运动方程的精确解,利用弹性点支处位移和反力的线性关系导出频率方程.最后以受弹性点支的圆膜为例给出了其频率方程的具体计算公式,并数值计算了受两个对称弹性点支的圆膜的固有振动频率.     

Estimation of Circle Parameters by Centroiding

In computer vision and computer aided manufacturing, it is often necessary to fit a circular arc through a number of noisy points. Determining the arc center and radius from a set of points is inherently a nonlinear problem and all estimators will exhibit the so-called threshold phenomenon. A combination of a short arc, small number of points, and large noise magnitude will create a threshold region (THR) whereby the estimation errors are several times larger than those above the THR. The transition into the THR is sudden. It is also difficult to determine the THR for an estimator. This paper presents an estimation scheme for the circle parameters by first computing different centers from all combinations of N data points, taken three at a time. A weighted average of those centers gives the final estimate. The procedure is simple, noniterative and simulation results show that it has a smaller THR than an estimator which is near-optimal when not operating inside the THR.     

Mathematical analysis of Zienkiewicz—Zhu's derivative patch recovery technique

Zienkiewicz-Zhu's derivative patch recovery technique is analyzed for general quadrilateral finite elements. Under certain regular conditions on the meshes, the arithmetic mean of the absolute error of the recovered gradient at the nodal points is superconvergent for the second-order elliptic operators. For rectangular meshes and the Laplacian, the recovered gradient is superconvergent in the maximum norm at the nodal points. Furthermore, it is proved for a model two-point boundary-value problem that the recovery technique results in an “ultra-convergent” derivative recovery at the nodal points for quadratic finite elements when uniform meshes are used. © 1996 John Wiley & Sons, Inc.     

On one atypical scheme of application of the second Lyapunov method

The second Lyapunov method is applied to the analysis of stability of triangular libration points in a three-dimensional restricted circular three-body problem. It is shown that the triangular libration points are unstable.     

Absolute points of correlations of $$PG(3,q^n)$$ P G ( 3 , q n )

Journal of Algebraic Combinatorics - The sets of the absolute points of (possibly degenerate) polarities of a projective space are well known. The sets of the absolute points of (possibly...     

Absolute points of correlations of $$PG(4,q^n)$$

Journal of Algebraic Combinatorics - The sets of the absolute points of (possibly degenerate) polarities of a projective space are well known. The sets of the absolute points of (possibly...     

On the minimal convex annulus of a planar convex body

In this paper we introduce the notion of a minimal convex annulusK (C) of a convex bodyC, generalizing the concept of a minimal circular annulus. Then we prove the existence — as for the minimal circular annulus — of a Radon partition of the set of contact points of the boundaries ofK (C) andC. Subsequently, the uniqueness ofK (C) is shown. Finally, it is proven that, for typicalC, the boundary ofC has precisely two points in common with each component of the boundary ofK (C).     

Ball-Polyhedra

We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other objects of study are bodies obtained as intersections of finitely many balls of the same radius, called ball-polyhedra. We find analogues of several results on convex polyhedral sets for ball-polyhedra.     

Polaritäten Ebener projektiver Ebenen

A projective plane is called flat if the spaces of points and lines are locally compact and 2-dimensional and the joining of points and the intersecting of lines are continuous. H. Salzmann studied planes of this type in [11]–[21]. Here polarities of such planes are considered. In II general properties of polarities of flat planes are discussed. For example, a polarity with absolute points has always an oval of absolute points. A flat projective plane with a cartesian ternary field K admits a polarity iff multiplication in K is commutative. In III the polarities of flat projective planes with a 3-dimensional collineation group are determined.     

On the total absolute curvature of closed curves in spheres

The total absolute curvature of a closed curve in a Euclidean space is always greater or equal to 2 (Fenchel's inequality,1929, [3], [1], [13]); especially for a knotted curve it is always greater than 4 (Fary-Milnor's inequality, 1949, [2], [7], [5], [4]).For the total absolute curvature of closed curves in spheres no such lower bounds exist because there are closed geodesies. Here we derive similar bounds which depend on the length of the curve resp.the area of surfaces of disk-type bounded by the curve.In order to prove these inequalities we start from the computation of the total absolute curvature as mean value of the number of critical points of certain level functions ([11],[12]); we use some topological considerations and Poincaré's integralgeometric formula for the computation of length resp. area.     

Convex polyhedra of doubly stochastic matrices—IV

Basic geometrical properties of general convex polyhedra of doubly stochastic matrices are investigated. The faces of such polyhedra are characterized, and their dimensions and facets are determined. A connection between bounded faces of doubly stochastic polyhedra and faces of transportation polytopes is established, and it is shown that there exists an absolute bound for the number of extreme points of d-dimensional bounded faces of these polyhedra.     

Gradient-Constrained Minimum Networks. III. Fixed Topology

The gradient-constrained Steiner tree problem asks for a shortest total length network interconnecting a given set of points in 3-space, where the length of each edge of the network is determined by embedding it as a curve with absolute gradient no more than a given positive value m, and the network may contain additional nodes known as Steiner points. We study the problem for a fixed topology, and show that, apart from a few easily classified exceptions, if the positions of the Steiner points are such that the tree is not minimum for the given topology, then there exists a length reducing perturbation that moves exactly 1 or 2 Steiner points. In the conclusion, we discuss the application of this work to a heuristic algorithm for solving the global problem (across all topologies).     

机器人圆形焊缝三维定位方法

自动化寻迹焊接过程中有许多焊缝为圆形,在焊缝特征点的坐标确定以后如何确定圆形焊缝的中心轨迹线是至关重要的.利用空间解析几何的原理,结合最小二乘法,既利用焊缝的已知信息又利用了实时监测点的信息,给出了圆形焊缝三维空间中心位置的确定方法.实验表明,该方法具有较高的精度和较强的实用性.     

Circle orders and angle orders

A poset is a circle order if its points can be mapped into circular disks in the plane so that x in the poset precisely when x's circular disk is properly included in y's; the poset is an angle order if its points can be mapped into unbounded angular regions that preserve < by proper inclusion. It is well known that many finite angle orders are not circle orders, but has been open as to whether every finite circle order is an angle order. This paper proves that there are finite circle orders that are not angle orders.     

Approximation of a planar cubic Bézier spiral by circular arcs

A planar cubic Bézier curve that is a spiral, i.e., its curvature varies monotonically, does not have internal cusps, loops, and inflection points. It is suitable as a design tool for applications in which fair curves are important. Since it is polynomial, it can be conveniently incorporated in CAD systems that are based on B-splines, Bézier curves, or NURBS. When machining objects, it is desirable that as much as possible of a curved toolpath be approximated by a sequence of circular arcs rather than straight-line segments. Such an arc-spline approximation of a planar cubic Bézier spiral is presented.     

k个内点支承圆板的对称弯曲

本文研究在均布载荷作用下,表面上有k个等距离内点支承的弹性圆板的对称弯曲.将自由边界位移和转角展开成Fourier级数,应用文献[6]的方法,使平衡方程和边界条件同时得到精确满足,从而获得了挠曲面方程的解析表达式.这是一种简便、有效的算法.