Line Congruence Research Articles

Interactive Visualization of Line Congruences for Spatial Mechanism Design

This paper presents a framework for generating, representing, and interacting with the line congruences associated with four general finite poses. These line congruences are the solution space of spatial 4C mechanisms which will guide a moving body through the four prescribed poses. Hence, the contributions of this paper are applicable to developing interactive tools for designing spatial 4C mechanisms for four pose motion generation. Moreover, the strategies employed to address this difficult interactive visualization challenge are presented. The goal here being to facilitate future works which address other interactive visualization challenges. First, a methodology for generating a parameterized representation of the line congruences is reviewed. This is followed by strategies for visually representing the line congruences which are appropriate for both workstation and immersive virtual reality computer graphics. Next, strategies and supporting algorithms for interacting with the line congruences to obtain solution mechanisms with fixed links or coupler links in desired regions of the workspace are presented. The result is an intuitive interactive visual design methodology for generating and interacting with the line congruences associated with four general finite spatial poses for spatial 4C mechanism design. It is our desire that this effort, albeit focused upon the challenge of creating computer-aided design environments for spatial 4C mechanisms, will facilitate as well as motivate other efforts to address the inherent visualization and interaction challenges in designing three dimensional mechanical systems.

Line Congruences of Low Order

We present a survey on congruences of lines of low order. We start by recalling the main properties of the focal locus, and use it to reobtain the classification of congruences of order one in P 3 . We then explain the main ideas of a work in progress with S. Verra, outlining how to classify congruences of lines of order two in P 3 . We end by stating the main problems on this topic.

Systems of conservation laws of Temple class, equations of associativity and linear¶congruences in P 4

We propose a geometric correspondence between (a) linearly degenerate systems of conservation laws with rectilinear rarefaction curves and (b) congruences of lines in projective space whose developable surfaces are planar pencils of lines. We prove that in P4 such congruences are necessarily linear. Based on the results of Castelnuovo, the classification of three-component systems is obtained, revealing a close relationship of the problem with projective geometry of the Veronese variety V2⊂P5 and the theory of associativity equations of two-dimensional topological field theory.

On first order congruences of lines of ℙ4 with a fundamental curve

On first order congruences of lines of ℙ4 with a fundamental curve

The Mixed Curvatures on C N

The aim of the present paper is devoted to the investigation of some geometrical properties on the middle envelope in terms of the invariants of the third quadratic form of the normal line congruence CN . The mixed middle curvature and mixed curvature on CN are obtained in tenus of the Mean and Gauss curvatures of the surface of reference. Our study is considered as a continuation to Stephanidis ([1], [2], [3], [4], [5]). The technique adapted here is based on the methods of moving frames and their related exteriour forms [6] and [7].

A focus on focal surfaces

Congruences of lines in P3, i.e. two-dimensional families of lines, and their focal surfaces, have been a popular object of study in classical algebraic geometry. They have been considered recently by several authors as Arrondo, Goldstein, Sols, Verra. Aim of the paper under review is to study from the modern point of view the notions appearing in this context, so to prove in a rigorous way some classical results. More precisely, a congruence is a surface X G(1, 3), its focal locus F, in general a surface, is the branch locus of the natural projection qX from the incidence correspondence IX to P3. It results that a general line of X contains two foci counted with multiplicity, a line L whose points are all foci is called a focal line. In the paper under review the authors compute several invariants of the focal surface of a smooth congruence, e.g. its degree, class and sectional genus, and the degree of its nodal and cuspidal curves (assuming that these are the only 1-dimensional components of its singular locus). Then they study in detail some important examples of congruences, i.e. the bisecant lines of a smooth curve C in P3, the bitangents and the inflectional tangents of a smooth surface . In the first case they find that in general the focal surface F is all formed by focal lines, which are the stationary bisecants. In the other examples they find, among other results, that is a component of the focal surface with high multiplicity, and that at least one component of F is formed by focal lines.Motivated by the examples they state a series of conjectures about congruences whose focal surface is not irreducible or not reduced, and about singularities of congruences of bitangents or inflectionary tangents to not necessarily smooth surfaces of P3.

Die Fl�chen mit geraden Linien als Affinen Zentrafl�chen

If the (euclidean) Gauss curvature of a surface in 3-space is nowhere vanishing, we have a uniqueaffine normal in every point of the surface and the set of affine normals forms a line congruence. According to the euclidean situation one can discuss existence and degeneration ofaffine focal surfaces. A cylinder of revolution in euclidean space can be characterized by the property, that all normals intersecttwo straight lines. The corresponding property in affine geometry leads to certainaffine surfaces of revolution, whose meridian curves (plane sections through a fixed axis) can be determined. If we assume the affine focal surfaces to coincide (the affine Weingarten endomorphism has double eigenvalues in every point) and degenerate intoone straight line, then the surface is aruled surface the generators of which areparallel to a plane.

Systems of conservation laws within the framework of the projective theory of congruences: the Lévy transformations of semi-Hamiltonian systems

We continue the investigation of the recently proposed geometric correspondence between systems of conservation laws and congruences of lines in projective space. The relationship between the `additional' conservation laws and hypersurfaces conjugate to a congruence is established, thus providing a basis for the Lévy transformations of semi-Hamiltonian systems. Similarly, the correspondence between commuting flows and certain families of planes (containing the lines of the congruence) gives rise to the adjoint Lévy transformations.

On the geometric origin of Orowan-type kinematic relations and the Schmid yield criterion

It is shown that the Orowan-type kinematic relations as well as the Schmid yield criterion can be derived basing oneself on a formula defining the mean curvature of glide surfaces for a principal congruence of edge dislocation lines accompanied with a particular distribution of secondary point defects. Moreover, it appears that this mean curvature has the physical meaning of a mesoscopic material parameter defining a relation between the evolution of the dislocation state and plastic deformation. It is pointed out that the existence of Orowan-type relations puts kinematic constraints, dependent on the isometry group of glide surfaces, on the dislocation density tensor.

Line congruences as surfaces in the space of lines

The space of lines in R 3 can be viewed as a four dimensional homogeneous space of the group of Euclidean motions, E(3). Line congruences arise in the classical method of transforming one surface to another by lines. These transformations are particularly interesting if some geometric property of the original surface is preserved. Line congruences, then, are two parameter families of lines and can be studied as surfaces in the space of lines. In this paper, we use the method of moving frames to study line congruences. We calculate the first order invariants of line congruences for which there are two real focal surfaces, and give the geometric meaning of these invariants. We look specifically at the case where the two first order invariants are constant and give a simple proof of Bäcklund's Theorem which relates to the transformation of one pseudospherical surface, a surface of constant negative Gaussian curvature, to another. These transformations are of interest since pseudospherical surfaces correspond to solutions to the sine-Gordon equation. We also give a proof of Bianchi's permutability theorem for pseudospherical surfaces in this context. Finally, we use the results of these theorems to generate some pseudospherical surfaces. All of these concepts and results are understood in terms of the structure equations of the line congruence.

Darboux Transformations Between Δα=sinh α and Δα=sin α and the Application to Pseudo-Spherical Congruences in R 2,1

In this Letter, it is proved that by using Darboux transformation, from a known solution of the sinh-Laplace equation (resp. sine-Laplace equation), new solutions of sine-Laplace equation (resp. sinh-Laplace equation) are obtained explicitly. The corresponding geometrical configuration is the space-like pseudo-spherical line congruence with two focal surfaces of negative constant curvature being space-like and time-like, respectively.

The Tangent Conoids Family Which Depends on the Ruled Surface

In this study, a new congruence[A••] has been definedby putting a tangent right conoid on each line of a ruled surface (A1(s)) of a line congruence [A]. Then, by considering special case of the congruence [A••] which has been defined in the previous part, the concepts of tangent congruence, drall and the relation among Blaschke vectors of Blaschke trihedrons, having common line Ao has been examined for this special case. At the end of this study, the concept of tangent congruence for some special congruences has been examined.

Depudecin makes a debut.

Although first identified as the product of sour grapes (acetum, vinegar), acetate recently has proven to be a sweet harvest for biologists interested in transcriptional regulation. Over the past few years, a congruence of many separate lines of investigation has revealed that acetylation of chromatin is likely to play a key role in gene regulation in eukaryotes, from yeast to mammals (reviewed in refs. 1–4). Indeed, acetate appears to serve as a molecular fulcrum by which chromatin structure is levered open and closed by transcription factors and thereby may contribute to the changes in gene expression that distinguish neoplastic from normal cells. Intriguingly, the crucial role of acetate in gene regulation may have been exploited by microorganisms long before these processes were evident to researchers: a number of microbial products appear to target the acetylation/deacetylation pathways that are critical for proper gene regulation and normal cell function (5).

Appendix to“congruences of small degree in G(1,4)”: pfaffian linkage in codimension three and applications to congruences

We introduce the notion of Pfaffian linkage in codimension three and give sufficient conditions for the linked variety to be smooth. As a result, we are able to construct smooth congruences of lines in P4 whose existence was an open problem.

On special line congruences and their Klein images on the absolutum of the 5-dimensional elliptic space

In this article special line congruences are studied in the 3-dimensional elliptic space and their Klein images on the absolutum of the 5-dimensional elliptic space . Also the inverse Klein images are studied. The methods adapted here are based on the Cartan’s methods of moving frames ([1], [2]).

Spreads admitting elliptic regulizations

Throughout this paper, the underlying projective space is 3-dimensional and Pappian. A spreadL admits aregulization ∑, if ∑ is a collection of reguli contained inL and if each element ofL, except at most two lines, is contained either in exactly one regulus of ∑ or in all reguli of ∑. Replacement of each regulus of ∑ by its complementary regulus (exceptional lines remain unchanged) produces thecomplementry congruence L ∑ of L with respect to ∑. IfL ∑ is an elliptic linear congruence of lines, then we call ∑ anelliptic regulization. Applying a method due to Thas and Walker we construct topological spreads of PG(3,ℝ) which admit one elliptic and no further regulization. For each of these spreads we determine the group of automorphic collineations. Among others we obtain also spreads which are the complete intersection of a general linear complex of lines and of a cubic complex of lines.

Laplace transformations of hydrodynamic-type systems in Riemann invariants: Periodic sequences

The conserved densities of hydrodynamic type system in Riemann invariants satisfy a system of linear second order partial differential equations. For linear systems of this type Darboux introduced Laplace transformations, generalising the classical transformations in the scalar case. It is demonstrated that Laplace transformations can be pulled back to the transformations of the corresponding hydrodynamic type systems. We discuss periodic Laplace sequences of with the emphasize on the simplest nontrivial case of period 2. For 3-component systems in Riemann invariants a complete discription of closed quadruples is proposed. They turn to be related to a special quadratic reduction of the (2+1)-dimensional 3-wave system which can be reduced to a triple of pairwize commuting Monge-Ampere equations. In terms of the Lame and rotation coefficients Laplace transformations have a natural interpretation as the symmetries of the Dirac operator, associated with the (2+1)-dimensional n-wave system. The 2-component Laplace transformations can be interpreted also as the symmetries of the (2+1)-dimensional integrable equations of Davey-Stewartson type. Laplace transformations of hydrodynamic type systems originate from a canonical geometric correspondence between systems of conservation laws and line congruences in projective space.

Passing a Central Axis Congruence Through Six Arbitrary Lines in Space

The design theory of a spatial 4C linkage to move a rigid body through four specified spatial positions determines a congruence of lines that may be used as fixed axes of the linkage, called the central axis congruence. In this paper, we locate a central axis congruence in space by identifying six arbitrary lines that it is to contain; we do not specify the positions of a rigid body or the associated relative displacement screws. We find that this yields at least one and as many as five different central axis congruences through a given set of six lines. Having defined the congruence without identifying any positions, we then show how to determine sets of four spatial positions that generate the given central axis congruence.

Spreads admitting net generating regulizations

We say a spread S carries a regulization Σ, if Σ is a collection of reguli contained in S and if each element of S, except at most two lines, is contained either in exactly one regulus of Σ or in all reguli of Σ. Replacement of each regulus of Σ by its complementary regulus (exceptional lines remain unchanged) yields the complementary congruence Sc Σ of S with respect to Σ. If Sc Σ is a hyperbolic or parabolic or elliptic linear congruence of lines, then Σ is called a net generating, in particular, a hyperbolic or parabolic or elliptic regulization, respectively. For hyperbolic and parabolic regulizations we also give other geometric characterizations.

The critical line congruence for reconstruction from three images

Let three projected images of a set of lines in space be given. Then, in general, the relative positions of the lines can be reconstructed uniquely up to a collineation of space. Reconstruction fails to be unique for certain critical sets of lines. It is known that each critical set is parameterised by a Bordiga surface in ℙ4. A new proof of this result is given. In addition, it is shown that every Bordiga surface parameterises a critical set of lines. The proof involves an explicit construction of the second or spurious set of lines which projects down to the same three images as the veridical set of lines.