A study on a line congruence as surface in the space of lines

<abstract><p>In this work, we introduce a line congruence as surface in the space of lines in terms of the E. Study map. This provides the ability to derive some formulae of surfaces theory into line spaces. In addition, the well known equation of the Plucker's conoid has been obtained and its kinematic-geometry are examined in details. At last, an example of application is investigated and explained in detail.</p></abstract>

On the one-parameter Lorentzian spatial motions

In this paper, differential properties of the one-parameter Lorentzian spatial motions are developed with explicit expressions independent of coordinates systems. In term of this, we calculate the Disteli formulae of a spacelike line trajectory and derive the connections with kinematic geometry of the axodes. Lastly, a theoretical expression of a spacelike inflection line congruence are investigated in detail.

Time-like line congruence in the dual Lorentzian 3-space 𝔻13

In this paper, we introduce a time-like line congruence which has the parameter ruled surfaces as principal ruled surfaces. Then we extend several well-known formulae of line congruence in the Euclidean 3-space to the Minkowski 3-space. Meanwhile, we report some geometric descriptions of these formulae. Finally, an example of application is introduced and described in detail.

Complexes of elliptic cylinders with a characteristic manifold of the generator element in the form of coordinate straight lines

The complex (three-parameter family) of elliptic cylinders is investigated in the three-dimensional affine space, in which the characteristic multiplicity of the forming element consists of three coordinate axes. The focal variety of the forming element of the considered variety is geometrically characterized. Geometric properties of the complex under study were obtained. It is shown that the studied manifold exists and is determined by a completely integrable system of differential equations. It is proved that the focal variety of the forming element of the complex consists of four geometrically characterized points. The center of the ray of the straight-line congruence of the axes of the cylinder, the indicatrix of the second coordinate vector, the second coordinate line and one of the coordinate planes are fixed. The indicatrix of the first coordinate vector describes a one-parameter family of lines with tangents parallel to the second coordinate vector. The end of the first coordinate vector describes a one-parameter family of lines with tangents parallel to the third coordinate vector. The indicatrix of the third coordinate vector and its end describe congruences of planes parallel to the first coordinate plane. The points of the first coordinate line and the first coordinate plane describe one-parameter families of planes parallel to the coordinate plane indicated above.

Synthesis of Equations For Ruled Surfaces With Two Curvilinear And One Rectangular Directrixes

Ruled surfaces have long been known and are widely used in construction, architecture, design and engineering. And if from the technical point of view the developable surfaces are more attractive, then architecture and design successfully experiment with non-developable ones. In this paper are considered non-developable ruled surfaces with three generators, two of which are curvilinear ones. According to classification, such surfaces are called twice oblique cylindroids. In this paper has been proposed an approach for obtaining of twice oblique cylindroids by immersing a curve in a line congruence of hyperbolic type. Real directrixes of such congruence are a straight line and a curve. It has been proposed to use helical lines (cylindrical and conical ones) as a curvilinear directrix, and a helical line’s axis as the straight one. Then the congruence’s rectilinear ray will simultaneously intersect the helical line and its axis. Congruence parameters are the line’s pitch and the guide cylinder or cone’s radius. The choice of the curvilinear directrix is justified by the fact that the helical lines have found a wide application in engineering and architecture. Accordingly, the helical lines based surfaces can have a great potential. In this paper have been presented parametric equations of the considered congruences. The congruence equations have been considered from the point of view related to introducing a new curvilinear coordinate system. The obtained system’s coordinate surfaces and coordinate lines have been also studied in the paper. To extract the surface, it is necessary to immerse the curve in the congruence. To synthesize the equations has been used a constructive-parametric method based on the substitution of the immersed line’s parametric equations in the congruence equations according to a special algorithm. In the paper have been presented 5 examples for the synthesis of ruled surfaces equations such as the twice oblique cylindroid and their visualization. The method is universal and algorithmic, and therefore easily adaptable for the automated construction of surfaces with variable parameters of both the congruence and the immersed line.

Forcing, damping and detuning for single and coupled Van der Pol oscillators

We use Melnikov function techniques together with geometric methods of bifurcation theory to study the interactions of forcing, damping and detuning on resonant periodic orbits for single and coupled forced Van der Pol oscillators. For a coupled pair the local bifurcation geometry is almost everywhere described in terms of the singularities of a line congruence in three dimensions.