Pencil Of Lines Research Articles

A Note on the Maximum Circle Inverses of Lines in the Maximum Plane

In this study, the images of lines under inversion in maximum circle are analytically examined. It is observed that the image of line not passing through the center of the inversion is not a maximum circle, but the closed curve containing at least one parabola arc. Some properties regarding the images of lines are introduced. Then, the images of line segments are examined according to the positions of their end points . In addition, it is seen that the inversion in maximum circle transforms the pencil of parallel lines (except line passing the center) to the set of the closed curves passing the center of inversion. Also, the images of the concurrent lines under inversion with respect to a maximum circle are considered and the results are presented.

Gronwall's conjecture for 3-webs with two pencils of lines

We prove the old-standing Gronwall conjecture in the particular case of linear 3-webs whose 2 foliations are 2 pencils of lines. For a non-hexagonal 3-web, we also introduce a family of projective torsion-free Cartan connections, the web leaves being geodesics for each member of the family, and give a web linearization criterion.

Remarks on Rigidity Properties of Conics

Inspired by the recent results toward Birkhoff conjecture (a rigidity property of billiards in ellipses), we discuss two rigidity properties of conics. The first one concerns symmetries of an analog of polar duality associated with an oval, and the second concerns properties of the circle map associated with an oval and two pencils of lines.

Constructions and Comparisons of Pooling Matrices for Pooled Testing of COVID-19.

In comparison with individual testing, group testing is more efficient in reducing the number of tests and potentially leading to tremendous cost reduction. There are two key elements in a group testing technique: (i) the pooling matrix that directs samples to be pooled into groups, and (ii) the decoding algorithm that uses the group test results to reconstruct the status of each sample. In this paper, we propose a new family of pooling matrices from packing the pencil of lines (PPoL) in a finite projective plane. We compare their performance with various pooling matrices proposed in the literature, including 2D-pooling, P-BEST, and Tapestry, using the two-stage definite defectives (DD) decoding algorithm. By conducting extensive simulations for a range of prevalence rates up to 5%, our numerical results show that there is no pooling matrix with the lowest relative cost in the whole range of the prevalence rates. To optimize the performance, one should choose the right pooling matrix, depending on the prevalence rate. The family of PPoL matrices can dynamically adjust their construction parameters according to the prevalence rates and could be a better alternative than using a fixed pooling matrix.

Fréchet-valued meromorphic extension along a pencil of complex lines

The aim of paper is to find the condition under which a Fréchet-valued function [Formula: see text] admitting meromorphic extension along some pencil of complex lines can be meromorphically extended to a neighborhood of [Formula: see text] Some auxiliary results concerning the domains of existence for Fréchet-valued meromorphic functions, Rothstein’s theorem, Levi extension theorem for meromorphic functions with values in a locally complete space, convergence of formal power series of Fréchet-valued homogeneous polynomials are also proved in this work.

Improved bounds for pencils of lines

We consider a question raised by Rudnev: given four pencils of $n$ concurrent lines in $\mathbb R^2$, with the four centres of the pencils non-collinear, what is the maximum possible size of the set of points where four lines meet? Our main result states that the number of such points is $O(n^{11/6})$, improving a result of Chang and Solymosi. We also consider constructions for this problem. Alon, Ruzsa and Solymosi constructed an arrangement of four non-collinear $n$-pencils which determine $\Omega(n^{3/2})$ four-rich points. We give a construction to show that this is not tight, improving this lower bound by a logarithmic factor. We also give a construction of a set of $m$ $n$-pencils, whose centres are in general position, that determine $\Omega_m(n^{3/2})$ $m$-rich points.

Geometry on the lines of polar spine spaces

The concept of spine geometry over a polar Grassmann space is introduced. We determine conditions under which the structure of lines together with a binary coplanarity relation is a sufficient system of primitive notions for polar spine spaces. These assumptions also allow to characterize polar spine spaces in terms of the binary relation of being in one pencil of lines.

Cyclic order: a geometric analysis

Concepts of order play an important role in many branches of mathematics. We start the article with an analysis of the notion of cyclic order in algebraic structures, which includes a characterization of cyclically ordered groups by cyclic cones and the introduction of the notion of a cyclically ordered field. We then study the role of cyclic order in the foundations of geometry. In Euclidean and in absolute geometry, order structures are introduced by linear orders (see Hilbert in Grundlagen der Geometrie, Teubner, Stuttgart, 1972; Coxeter in Introduction to geometry, Wiley, New York, 1961; Sperner in Beziehungen zwischen geometrischer und algebraischer Anordnung. Sitzungsberichte der Heidelberger Akademie der Wissenschaften, 1949; Bachmann in Aufbau der Geometrie aus dem Spiegelungsbegriff, Springer, Heidelberg, 1973; H Struve and R Struve in J Geom 105:419–447, 2014; R Struve in J Geom 106:551–570, 2015). This excludes elliptic geometry. We show that the notion of cyclic order (on pencils of lines) allows the introduction of order structures in a unified way (including the elliptic case) and corresponds on the algebraic side to a linear order of the associated coordinate field. In addition we prove that the three classical geometries (Euclidean, hyperbolic, and elliptic) over fields K of characteristic $$\ne 2$$ are orderable if and only if a separation relation on rows of collinear points and a separation relation on pencils of concurrent lines can be defined which are ‘compatible’. The article closes with a geometric interpretation of cyclically ordered fields as $$\hbox {Gau}\ss \hbox {ian}$$ coordinate fields of Euclidean Hilbert planes.

Testing Hyperbolicity of Real Polynomials

Hyperbolic polynomials are real multivariate polynomials with only real roots along a fixed pencil of lines. Testing whether a given polynomial is hyperbolic is a difficult task in general. We examine different ways of translating hyperbolicity into nonnegativity conditions, which can then be tested via sum-of-squares relaxations.

Аппроксимация множеств прямых на плоскости

A few general lines in the ordinary Euclidean plane are said to be line generators of a plane linear set. To be able to say that every line of the set belongs to one-parametrical line set we have to find their envelope. We thus create a pencil of lines. In this article it will be shown that there are a finite number of pencils in one linear set. To find a pencil of lines the linear parametrical approximation is applied. Almost all of problems concerning the parametrical approximation of figure sets are well known and deeply developed for any point sets. The problem of approximation for non-point sets is an actual one. The aim of this paper is to give a path to parametrical approximation of linear sets defined in plane. The sets are discrete and consist of finite number of lines without any order. Each line of the set is given as y = ax + b. Parametrical approximation means a transformation the discrete set of lines into completely continuous family of lines. There are some problems. 1. The problem of order. It is necessary to represent the chaotic set of lines as well-ordered one. The problem is solved by means of directed circuits. Any of chaotic sets has a finite number of directed circuits. To create an order means to find all directed circuits in the given set. 2. The problem of choice. In order to find the best approximation, for example, the simplest one it is necessary to choose the simplest circuit. Some criteria of the choice are discussed in the paper. 3. Interpolation the set of line factors. A direct approach would simply construct an interpolation for all line factors. But this can lead to undesirable oscillations of the line family. To eliminate the oscillations the special factor interpolation are suggested. There are linear sets having one or several multiple points, one or several multiple lines and various combinations of multiple points and lines. Some theorems applied to these cases are formulated in the paper.

Мнимые точки в декартовой системе координат

Geometric models are considered that allow symbolic representation of imaginary points on a real Cartesian coordinate plane XY. The models are based on the fact that through every pair of imaginary conjugate points A~B with complex coordinates x = a ± jb, y = c ± jd one unique real line m passes. For the image of imaginary points, it is proposed to use the graphic symbol m{OL} consisting of the line m passing through the imaginary points, the center O of the elliptic involution σ with imaginary double points A~B on the line m, and the Laguerre point L, from which the corresponding points involutions σ are projected by an orthogonal pencil of lines. According to A.G. Hirsch, the symbol m{OL} is called the marker of imaginary conjugate points A~B. A theorem is proved that establishes a one-to-one correspondence between the real Cartesian coordinates of the points O, L of the marker, and the complex Cartesian coordinates of the pair of imaginary conjugate points represented by this marker. The proved theorem allows us to solve both the direct problem (the construction of a marker depicting these imaginary points) and the inverse problem (the determination of the Cartesian coordinates of imaginary points represented by the marker). A graphical algorithm for constructing a circle passing through a real point and through a pair of imaginary conjugate points is proposed. An example of the graph-analytical determination of the Cartesian coordinates of imaginary points of intersection of two conics that have no common real points is considered.

On the Real Group Actions Preserving a Pencil of Straight Lines in the Lobachevskii Space

In the Beltrami-Klein model of the Lobachevskii space, we obtain explicit expressions for the real multiplicative group actions preserving hyperbolic pencil of straight lines, and for the real additive group actions preserving parabolic pencil of straight lines.

Length in the Cremona group

The Cremona group is the group of birational transformations of the plane. A birational transformation for which there exists a pencil of lines which is sent onto another pencil of lines is called a Jonqui\`eres transformation. By the famous Noether-Castelnuovo theorem, every birational transformation $f$ is a product of Jonqui\`eres transformations. The minimal number of factors in such a product will be called the length, and written $\mathrm{lgth}(f)$. Even if this length is rather unpredictable, we provide an explicit algorithm to compute it, which only depends on the multiplicities of the linear system of $f$. As an application of this computation, we give a few properties of the dynamical length of $f$ defined as the limit of the sequence $n \mapsto \mathrm{lgth} (f^n) / n$. It follows for example that an element of the Cremona group is distorted if and only if it is algebraic. The computation of the length may also be applied to the so called Wright complex associated with the Cremona group: This has been done recently by Lonjou. Moreover, we show that the restriction of the length to the automorphism group of the affine plane is the classical length of this latter group (the length coming from its amalgamated structure). In another direction, we compute the lengths and dynamical lengths of all monomial transformations, and of some Halphen transformations. Finally, we show that the length is a lower semicontinuous map on the Cremona group endowed with its Zariski topology.

A Novel Pencil Drawing Algorithm Based on Non-Symmetry and Anti-Packing Pattern Representation Model

The artistic stylization transformation of images is an important part in the research of Non-Photorealistic Rendering (NPR). This kind of non-realistic stylized research has a very strong realistic romantic temperament and it can express the artistic pursuit of researchers very well. It is a good example of practical application combining computer science and art. In the technology of image style conversion, the simulation of the realistic art style includes mainly cartoon style, oil painting style, ink painting style and pencil drawing style. Previously, some researchers proposed a style conversion method based on the prior knowledge that there might be crosses at the junction of two lines where the tone distribution has a certain rule. However, in this paper, we have focused on the use of an edge extraction algorithm based on Non-symmetry and Anti-packing pattern representation Model (NAM) to enhance the pencil strokes and highlight the details of the image. Then, by combining the line pencil strokes with the tone drawing, we can transform the image into an image with pencil sketch style. Finally, compared with the most important previous methods, the experimental results presented in this paper showed that our proposed method showed more clearly the scene and the content and more details of the image than the previous methods. We also verified our method and calculated Structural Similarity (SSIM) and Peak Signal-to-Noise Ratio (PSNR) of original images and color pencil drawings with different algorithms on the 7 nature images and the BSDS500 data set.

Angular Radon spectrum for rotation estimation

This paper presents a robust method for rotation estimation of planar point sets using the Angular Radon Spectrum (ARS). Given a Gaussian Mixture Model (GMM) representing the point distribution, the ARS is a continuous function derived from the Radon Transform of such distribution. The ARS characterizes the orientation of a point distribution by measuring its alignment w.r.t. a pencil of parallel lines. By exploting its translation and angular-shift invariance, the rotation angle between two point sets can be estimated through the correlation of the corresponding spectra. Beside its definition, the novel contributions of this paper include the efficient computation of the ARS and of the correlation function through their Fourier expansion, and a new algorithm for assessing the rotation between two point sets. Moreover, experiments with standard benchmark datasets assess the performance of the proposed algorithm and other state-of-the-art methods in presence of noisy and incomplete data.

Geometry on the lines of spine spaces

Spine spaces can be considered as fragments of a projective Grassmann space. We prove that the structure of lines together with a binary coplanarity relation, as well as with the binary relation of being in one pencil of lines, is a sufficient system of primitive notions for these geometries. It is also shown that, over a spine space, the geometry of pencils of lines can be reconstructed in terms of the two binary relations.

Pencilled regular parallelisms

Over any field \({\mathbb{K}}\), there is a bijection between regular spreads of the projective space \({{\rm PG}(3,\mathbb{K})}\) and 0-secant lines of the Klein quadric in \({{\rm PG}(5,\mathbb{K})}\). Under this bijection, regular parallelisms of \({{\rm PG}(3,\mathbb{K})}\) correspond to hyperflock determining line sets (hfd line sets) with respect to the Klein quadric. An hfd line set is defined to be pencilled if it is composed of pencils of lines. We present a construction of pencilled hfd line sets, which is then shown to determine all such sets. Based on these results, we describe the corresponding regular parallelisms. These are also termed as being pencilled. Any Clifford parallelism is regular and pencilled. From this, we derive necessary and sufficient algebraic conditions for the existence of pencilled hfd line sets.

Реконструкция квадратичного кремонова преобразования

The geometric correspondence between the points of two planes can be considered well defined only when base data for its establishing is available, and a construction method by which its possible on the basis of these data for each point in one plane to find the corresponding points in the other one. Quadratic Cremona transformation can be specified by pointing out in the combined plane seven pairs of corresponding points. Naturally there is a need to establish a method for constructing any number of corresponding points. An outstanding Russian geometer K.A. Andreev indicated the linear construction based on the consideration of two correlations by which for each eighth point in the one plane is found the corresponding point of the other one. But in his work was not set up a problem to construct excluded (fundamental) points of quadratic Cremona transformation specified by seven pairs of points. There are many constructive ways to obtain the quadratic transformation in the plane. For example, it can be obtained by using two pairs of projective pencils of straight lines with vertices at the fundamental points (F-points). K.A. Andreev noted that this method for establishing of quadratic correspondence spread only to those cases when all F-points are the real ones. This statement is true for the 19th century’s level of geometric science, but today it’s too categorical. The theory of imaginary elements in geometry allows to develop a universal algorithm for construction of corresponding points in a quadratic transformation, given both by real and imaginary F-points. Summarizing the K.A. Andreev task, we come to the problem of finding the fundamental points (F-points) for a quadratic transformation specified by seven pairs of corresponding points. Almost one and half century the K.A. Andreev generalized task remained unsolved. The formation of this task’s constructive solution algorithm and its practical implementation has become possible by means of modern computer geometric modeling. According to proposed algorithm, the construction of F-points is reduced to the construction of second order auxiliary curves, on which intersection are marked the required F-points. The result received in this paper is used for development of the Cremona transformations’ theory, and for further application of this theory in the practice of geometric modeling.

A note on Clifford parallelisms in characteristic two

It is well known that a purely inseparable field extension L/F with some extra property and degree [L : F] = 4 determines a Clifford parallelism on the set of lines of the three-dimensional projective space over F. By extending the ground field of this space from F to L, we establish the following geometric description of such a parallelism in terms of a distinguished ‘absolute pencil of lines’ of the extended space: Two lines are Clifford parallel if, and only if, there exists a line of the absolute pencil that meets both of them. Mathematics Subject Classification (2010): 51A15, 51A40, 51J10, 51J15.

Rational maps of ℙn with prescribed fixed points and the smooth conic case

We construct rational maps of ℙn which have a prescribed variety as a component of their fixed point set. The resulting maps fix a pencil of lines for the case of hypersurfaces; thus including the cases of plane curves. We also determine the Cremona maps among the constructed ones for quadratic hypersurfaces. Our methods are based on associated matrices of forms of constant degree and the "triple action" of G = PGL n+1 on them. We include a complete classification of these maps and matrices for the case of the smooth conic curve in ℙ2. We obtain invariants and canonical forms for the orbits of our matrices under the triple action of G, modulo syzygies of a row vector. We obtain invariants and canonical forms for the orbits of the constructed rational maps under conjugation by G.