Real Straight Lines Research Articles

Мнимые прямые в декартовой системе координат

A geometric model of imaginary conjugate straight lines a~b, allowing symbolic representation of these lines on the real coordinate plane xy is considered. In order to connect the algebraic and geometric representations of imaginary straight lines, it is proposed to use the “mark” formed by orthogonal d1 ⊥ d2 and main g1~g2 directions of the elliptic involution σ in the pencil V. The specification of two pairs of pulling apart each other real straight lines d1~d2, g1~g2 passing through V, uniquely defines the elliptic involution σ in the pencil V, therefore, the V(d1 ⊥ d2, g1~g2) mark completely defines a pair of imaginary double straight lines a~b of elliptic involution σ(V), that allows consider the mark as an “image” of these imaginary straight lines. When using a mark, it is required to establish a one-to-one correspondence between complex coefficients of imaginary double straight lines equations and a graphically given mark. The direct and inverse problems are solved in this paper. The direct one is creation a mark representing imaginary straight lines, given by its own equations. The inverse one is determination of coefficients for the equations of imaginary lines defined by the mark. The essence of the direct and inverse problems consists in establishing a oneto-one correspondence between the equations of imaginary double straight elliptic involutions σ in the pencil V, and a graphically given mark containing the orthogonal and main directions of this involution. To solve both the direct and inverse problems, the Hirsch theorem (A.G. Hirsch) is used, which establishes a one-to-one correspondence between the complex Cartesian coordinates for a pair of imaginary conjugated points and real coordinates of a special “marker” symbolically representing these points. Have been considered examples of solution for geometric problems involving imaginary lines. In particular, has been solved the problem of constructing a circle passing through a given point and touching imaginary lines defined by its mark V(d1 ⊥ d2, g1~g2). Has been proposed a graphical and analytical algorithm for determining the coefficients of equations of imaginary tangents, traced to a conic section from its inner point.

Обобщенные многообразия Кенмоцу постоянного типа

В работе мы рассматриваем обобщенные многообразия Кенмоцу, мы вводим четвертое и пятое фундаментальные тождества обобщенных многообразий Кенмоцу, вводятся первый и второй структурные тензоры обобщенных многообразий Кенмоцу и доказаны их свойства, вводится понятие присоединенной Q-алгебры для обобщенных многообразий Кенмоцу. Доказано, что обобщенное многообразие Кенмоцу, а также специальные обобщенные многообразия Кенмоцу II рода имеют антикоммутативную присоединенную Q-алгебру.А многообразия Кенмоцу и специальные обобщенные многообразия Кенмоцу I рода имеют абелеву присоединенную Q-алгебру. Вводится контактный аналог постоянства типа и подробно исследуются обобщенные многообразия Кенмоцу постоянного типа. Получены условия точечного постоянства типа обобщенных многообразий Кенмоцу на пространстве присоединенной G-структуры. Доказано, что класс GK-многообразий нулевого постоянного типа совпадает с классом многообразий Кенмоцу, а класс GK-многообразий ненулевого постоянного типа конциркулярным преобразованием переводит-ся в почти контактное метрическое многообразие локально эквивалентное произведению шестимерного собственногоNK-многообразия на вещественную прямую.

Features of the Process of Stretch Forming Shells Convex-Concave Shape

It is proposed to use actual computer software systems, providing a visualization and automation of the wrinkling and visualization and automation of the results of the numerical finite-element modeling of stretch forming a convex-concave shape in view of the fact that the saddle-backed surface of the shell has two families of asymptotic real straight lines. Along these lines, the saddle-backed surface of the shell with convex-concave shape is easy to break up or riffles are formed along them. The objectives of computer simulation are to predict unwanted distortion geometric form of the shell because of wrinkling and to find a ways to minimize the deviations of its surface deviations from the required shape.

RULED SPECIAL LAGRANGIAN 3-FOLDS IN [Copf ]3

This is the fifth in a series of papers constructing explicit examples of special Lagrangian submanifolds in ${\mathbb C}^m$. A submanifold of ${\mathbb C}^m$ is ruled if it is fibred by a family of real straight lines in ${\mathbb C}^m$. This paper studies ruled special Lagrangian 3-folds in ${\mathbb C}^3$, giving both general theory and families of examples. Our results are related to previous work of Harvey and Lawson, Borisenko, and Bryant. Special Lagrangian cones in ${\mathbb C}^3$ are automatically ruled, and each ruled special Lagrangian 3-fold is asymptotic to a unique special Lagrangian cone. We study the family of ruled special Lagrangian 3-folds N asymptotic to a fixed special Lagrangian cone N0. We find that this depends on solving a linear equation, so that the family of such N has the structure of a vector space. We also show that the intersection $\Sigma$ of N0 with the unit sphere ${\mathcal S}^5$ in ${\mathbb C}^3$ is a Riemann surface, and construct a ruled special Lagrangian 3-fold N asymptotic to N0 for each holomorphic vector field w on $\Sigma$. As corollaries of this we write down two large families of explicit special Lagrangian 3-folds in ${\mathbb C}^3$ depending on a holomorphic function on $\mathbb C$, which include many new examples of singularities of special Lagrangian 3-folds. We also show that each special Lagrangian T2-cone N0 can be extended to a 2-parameter family of ruled special Lagrangian 3-folds asymptotic to N0, and diffeomorphic to $T^2\times{\mathbb R}$.2000 Mathematical Subject Classification: 53C38, 53D12.

On Inequalities of the Landau–Kolmogorov–Hörmander Type on a Segment and Real Straight Line

We prove inequalities of the Landau–Kolmogorov–Hormander type for the uniform norms (on some subinterval) of positive and negative parts of intermediate derivatives of functions defined on a finite interval. By using the limit transition, we obtain a new proof or the well-known Hormander result.

Roots of algebraic equations and Clifford algebra

If an algebraic equation onIR admits real roots it admits hyperbolic roots which are elements of the set (x0 + ey0) where e is a Clifford number having a square equal to 1. The equation can have hyperbolic or real roots $${{(a_1 + a_2 )} \over 2} + \varepsilon {{(a_1 - a_2 )} \over 2}$$ ifa1 anda2 are two real roots obtained for e=1 or e=−1. Ifn is the degree of an equation the maximum number of hyperbolic or real roots isn2. We apply our analysis to the intersection of plane curves with a real straight line and to differential linear equations with constant coefficients. *** DIRECT SUPPORT *** A00PF004 00003

A Realistic Digitization Model of Straight Lines

Understanding digital lines comprises two components: (i) determining whether a given digital set is a digital line segment, i.e., whether there exists a real straight line in R2that digitizes to it; and (ii) recovering the space of straight lines which digitize to a given digital line segment. Although these problems have been extensively studied by researchers in computer vision and digital geometry, the primary digitization model that has been used is object boundary quantization which is not a realistic sensor model in computer vision. In this paper, we present a realistic mathematical model of a digitized edge which handles both blurring and arbitrary thresholding. We show that a thresholded digital image of a blurred half-plane obtained for some unknown threshold value is equal to the image of a perfectly focused half-plane with the same slope obtained by object boundary quantization. This result implies that recovering the slope of a blurred half-plane, given its image obtained for some unknown threshold value, reduces to recovering the slope of a perfectly focused half-plane under object boundary quantization. Therefore, the previous results and algorithms for the recovery of straight lines, which mainly assumed object boundary quantization, are also valid if we assume this more realistic digitization model.

Spin and pin − structures, immersed and embedded surfaces and a result of Segre on real cubic surfaces

THE initial motivation of this paper comes from a result of Segre [ 121 about the real lines on a real cubic surface. As it is well known a smooth complex cubic surface has exactly 27 (complex) lines. In the real case this is not always true anymore. A smooth real cubic surface can have 27,15,7 or 3 real lines. This has been well known since the 19th century. The result of Segre we are alluding to is far less known and introduces a more subtle difference between the real and complex cases. Segre distinguishes two types of real straight lines (see below Section 6 for precise definitions) and shows that on a real cubic surface with 27 real lines 15 are of one type and 12 of the other (in fact the result is more complete and gives the classification in all cases-see Theorem 6.2 below). Segre proved this result by studying the degeneration of non-singular cubic surfaces to the union of 3 planes and a special “graphical” way of representing the occurring situations. Noting that the basic difference between the two types of lines is that their respective tubular neighbourhoods in the surface differ by a full twist in P3, our initial aim was to give a new interpretation and a new proof of this result in terms of the Pinstructure induced by the embedding of the surface in P3 (p3 taken with a fixed Spin structure). More precisely, we will show that the two type of lines distinguished by Segre are also differentiated at the homology level by the mod4 quadratic form canonically associated with the above Pinstructure. A further point of interest is that, assuming that the complexification X(C) c p’(C) of the surface X is also non-singular and that the surface is an M-surface, there is another Pinstructure, induced by the embedding of X(R) in X(@) (see [6]). This second form differs from the first by a “privileged” class in H’ (X, Z/2), a class which seems to deserve further investigations. We will explicitly compute this class for quadric and cubic surfaces. The work done for surfaces in P3 led us to study more generally immersions of surfaces in arbitrary orientable 3-manifolds. Using the theory of Spin and Pinstructures (see, for example, [lo] and the book [6]), we consider the problem of associating, as above, quadratic forms with immersions of surfaces in 3-manifolds. We have done this by reformulating results of Pinkall [ 111, where only the case of Iw3 is considered and results of Hass and Hughes [S] where the immersions of surfaces into arbitrary 3-manifolds is studied, but not in terms of quadratic forms. Following Pinkall we will also introduce the notion of immersed surfaces (an equivalence class of immersions-see Sections 4 and 10) and study the relationships between different equivalence relations on immersed surfaces (regular homotoppy, cobordism, equivalence of the Pinstructures) extending the results Pinkall

A purely geometric representation of all points in the projective plane

The advantage of having a basis of Geometric reality for the complex numbers of Algebra has long been apparent to mathematicians. More than two centuries ago JOHN WALLIS t devised a method of representing imaginary numbers by points in a plane which is truly ingenious, although hopelessly inferior to the well-known method with which we habitually associate the names of ARGAND and GAUSS. t From another point of view, this latter method may be looked upon as a method of representing, by means of real points in a plane, the imaginary points of a real straight line. The first mathematician to give a satisfactory purely geometric definition of imaginary elements was VON STAUDT, and in his Beitrage zur Geornetrie der Lage ? he shows how the GAUSS construction may be reached from a totally different point of view. He does not however outline his method in its most general form, as he makes use of the circular points at infinity, and a better statement has been given by K6TTER. 11 The same construction was used by HENRY J. S. SMITH ?, but the end in view was different and he did not employ VON STAUDT'S distinction between conjugate imaginaries. The problem of representing by means of real elements all points, not of a real straight line merely, but of a real plane, has not, to my knowledge, been