Space E3 Research Articles

Sweeping surfaces with Natural mate curve of a spatial curve in Euclidean 3-Space

In this paper, we introduce the notion of sweeping surfaces with Natural mate curve of a spatial curve in Euclidean 3-space E3 . We also show that the parametric curves on these surfaces are lines of curvature. Then, we derive the necessary and sufficient condition for the sweeping surface to become a developable ruled surface. In particular, we analyze the necessary and sufficient conditions when the resulting developable surface is a cylinder, cone or tangent surface. Finally, some representative curves are chosen to construct the corresponding developable surfaces which possessing these curves as lines of curvature.

On the Determination of Ruled and Developable Surfaces in Euclidean 3-space E³

Depending on E. Study’s map, the method of determination of ruled and developable surfaces is introduced. We show that dual vectorial expression of ruled and developable surfaces will be gained from coordinates and the first derivatives of the base curve. We illustrated this method by giving some representative examples.

Timelike Sweeping Surfaces According to Type-2 Bishop Frame in Minkowski 3–space

In this paper, we express timelike sweeping surfaces using rotation minimizing frames in Minkowski 3–Space E3 1 . Necessary and sufficient conditions for timelike sweeping surfaces to be developable ruled surfaces are derived. Using these, we analyze the conditions when the resulting timelike developable surface is a cylinder, cone or tangential surface.

IDENTIFYING CANAL SURFACES IN E4 : CHARACTERIZATION OF A SPECIAL CASE

This study, develops a way for representation of canal surfaces in 4-dimensional Euclidean space E4 . It examines the fundamental forms, Gaussian and mean curvature for a special type canal surface. Moreover, the conditions of both Weingarten and linear Weingarten canal surfaces are given for this new special type. Finally, the graphs of the projections of the canal surfaces using different radius functions in E4 are presented.

Global Existence for Stochastic Strongly Dissipative Zakharov Equations

The stochastic strongly dissipative Zakharov equations with white noise are studied. On the basis of the time uniform a priori estimates, we prove the existence and uniqueness of solutions in energy spaces E1 and E2, by using the standard Galerkin approximation method of stochastic partial differential equations.

Existence of immovability lines of a partial mapping of Euclidean space E5

It is considered a set of smooth lines such that through a point X ∈ Ω passed one line of given set in domain Ω ⊂ E5. The moving frame is frame of Frenet for the line ω1 of the given set. Integral lines of the vector fields are formed net ∑5 of Frenet. There exists a point on the tangent of the line ∑5. When a point X is shifted in the domain Ω the point describes it’s domain in E5. It is defined the partial mapping , such that . Necessary and sufficient conditions of immovability and degeneration of lines and in partial mapping are obtained.

Cell‐based maximum entropy approximants for three‐dimensional domains: Application in large strain elastodynamics using the meshless total Lagrangian explicit dynamics method

SummaryWe present the cell‐based maximum entropy (CME) approximants in E3 space by constructing the smooth approximation distance function to polyhedral surfaces. CME is a meshfree approximation method combining the properties of the maximum entropy approximants and the compact support of element‐based interpolants. The method is evaluated in problems of large strain elastodynamics for three‐dimensional (3D) continua using the well‐established meshless total Lagrangian explicit dynamics method. The accuracy and efficiency of the method is assessed in several numerical examples in terms of computational time, accuracy in boundary conditions imposition, and strain energy density error. Due to the smoothness of CME basis functions, the numerical stability in explicit time integration is preserved for large time step. The challenging task of essential boundary condition (EBC) imposition in noninterpolating meshless methods (eg, moving least squares) is eliminated in CME due to the weak Kronecker‐delta property. The EBCs are imposed directly, similar to the finite element method. CME is proven a valuable alternative to other meshless and element‐based methods for large‐scale elastodynamics in 3D. A naive implementation of the CME approximants in E3 is available to download at https://www.mountris.org/software/mlab/cme.

The new study of some characterization of canal surfaces with Weingarten and linear Weingarten types according to Bishop frame

In this paper, we have a tendency to investigate a particular Weingarten and linear Weingarten varieties of canal surfaces according to Bishop frame in Euclidean 3-space E3 satisfying some fascinating and necessary equations in terms of the Gaussian curvature, the mean curvature, and therefore the second Gaussian curvature. On the premise of those equations, some canal surfaces are introduced.

On Classifications of Normal and Osculating Curves in 3-dimensional Sasakian Space

This study provides the de…nition of rectifying, normal and osculating curves in 3-dimensional Sasakian space with their characterizations. Furthermore, the di¤erential equations obtained from these characterizations are solved and their figures are presented in the text. Sasakian manifolds were introduced in 1960 by the Japenese geometer Shigeo Sasaki [19]. There was not much activity in this field after the mid-1970s, until the advent of String theory. Since then Sasakian manifolds have gained importance in physics and geometry. For physicists and geometers, the study of Sasakian space has its own interest, so it has been extensively studied area of scientific researchs [1, 3, 4, 7, 9, 20]. In the 3-dimensional Sasakian space, to each regular curve γ, it is possible to associate three mutually ortogonal unit vector fields. The vectors V1, V2, V3 are called the tangent, the principal normal and the binormal vector field, respectively. The planes spanned by the vector fields, {V1, V2} , {V1, V3} and {V2, V3} are defined as the osculating plane, the rectifying plane and the normal plane, respectively. In the Euclidean space E3, the notion of rectifying curves was introduced by Chen in [10]. In addition, he showed in [11] that there exist a simple relationship between the rectifying curves and centrodes, which play some important roles in mechanics, kinematics as well as differentialgeometry.

Diophantine Geometry in Space E2 and E3

In this paper, we are focusing on solving Diophantine equations with geometry. Integers that are solutions of Diophantine equations are represented in space by a lattice. We will use the fact that equations can be interpreted geometrically as curves or surfaces. To determine the properties of the solutions, we will use tangents (of the surface).

Surfaces family with a common geodesic curve in Euclidean 3-space E3

Surfaces family with a common geodesic curve in Euclidean 3-space E3

Normal Ruled Surfaces of a Surface along a Curve in Euclidean 3-Space E3

In this paper, we define a ruled surface normal to a surface along a curve on the surface. Then, we analyze the necessary and sufficient condition for that surface to be normal developable. Also, we solve the problem when the resulting developable surface is a cylinder, cone or tangent surface. Finally, we give some representative examples.

Some Null Quaternionic Curves in Minkowski spaces

In this work, we examine null quaternionic rectifying curves and null quaternionic similar curves in Minkowski space E1^3. Also, we defined null quaternionic (1,3)-Bertrand partner curves in E1^4. Thus, we have characterizations between curvatures of these curves in Minkowski spaces.

Conchoid curves and surfaces in Euclidean 3-Space

In this study firstly, we study with conchoid curves in Euclidean plane E2. We calculate the curvature of the conchoid curve and give some results. Furthermore, we consider the surface of revolution given with the conchoid curve in Euclidean 3-space E3. The Gaussian and mean curvature is calculated of these surfaces. Also we give some examples and plot their graphics. Finally we study conchoidal surface in Euclidean 3-space. We give some results for the conchoidal surface to become flat and minimal. We give an example and plot the garphics of the conchoidal surfaces.

Global well-posedness of the two-dimensional Benjamin equation in the energy space

In this paper, the global well-posedness of the Cauchy problem for the two-dimensional Benjamin equation ∂tu+∂x3u−ϵH∂x2u−∂x−1∂y2u+∂x(u2∕2)=0,u(x,y,0)=ϕ(x,y)in the energy space E1={u:‖u‖L2+‖∂xu‖L2+‖∂x−1∂yu‖L2<∞} is obtained.

A surface in W2,p is a locally Lipschitz-continuous function of its fundamental forms in W1,p and Lp, p > 2

The fundamental theorem of surface theory asserts that a surface in the three-dimensional Euclidean space E3 can be reconstructed from the knowledge of its two fundamental forms under the assumptions that their components are smooth enough—classically in the space C2(ω) for the first one and in the space C1(ω) for the second one—and satisfy the Gauss and Codazzi–Mainardi equations over a simply-connected open subset ω of R2; the surface is then uniquely determined up to proper isometries of E3. Then S. Mardare showed in 2005 that this result still holds under the much weaker assumptions that the components of the first form are only in the space Wloc1,p(ω) and those of the second form only in the space Llocp(ω), the components of the immersion defining the reconstructed surface being then in the space Wloc2,p(ω), p>2.The purpose of this paper is to complement this last result as follows. First, under the additional assumption that ω is bounded and has a Lipschitz-continuous boundary, we show that a similar existence and uniqueness theorem holds with the spaces Wm,p(ω) instead of Wlocm,p(ω). Second, we establish a nonlinear Korn inequality on a surface asserting that the distance in the W2,p(ω)-norm, p>2, between two given surfaces is bounded, at least locally, by the distance in the W1,p(ω)-norm between their first fundamental forms and the distance in the Lp(ω)-norm between their second fundamental forms. Third, we show that the mapping that uniquely defines in this fashion a surface up to proper isometries of E3 in terms of its two fundamental forms is locally Lipschitz-continuous.

Position Vectors of General Helices According to Type-2 Bishop Frame in E^3

In this paper, we study the position vector of a general helix according to type-2 Bishop frame in the 3-dimensional Euclidean space E3 . Moreover we determine the natural representation of a general helix in E^3 .

On a Torus Model

The surface of parallel transfer is a surface that admits the parametrization r(u,v) = U(u) + V(v). The surface of parallel transfer in E3 can be considered as a surface produced by parallel transfer of one line along the other.In this study the torus M is different from the classic torus T, which is obtained by rotating the circle along the axis. We consider the torus as the surface of parallel transfer obtained by parallel transfer of one circle along the other. The circles are located in mutually orthogonal intersecting planes.The closed curve on the torus M is defined with the 4n-periodic vector-function.Using this function, the equations of one-sided surfaces are obtained. In particular, the equation of the Mobius band with the mentioned curve being the boundary is found.The Klein bottle is cut into two Moebius band. Also, the cross-cap is considered.The inversion of torus T and the inversion of torus M is studied in this paper.The studied surfaces are constructed in Euclidean space E3 with the help of a mathematical software toolkit.DOI 10.14258/izvasu(2018)1-24

Estimate on the Dimension of Global Attractor for Nonlinear Higher-Order Coupled Kirchhoff Type Equations

In this paper, we investigate the finite dimensions of the global attractor for nonlinear higher-order coupled Kirchhoff type equations with strong linear damping in Hilbert spaces E0 and E1. Under the appropriate assumptions, we acquire a precise estimate of the upper bound for its Hausdorff and Fractal dimensions.

On biconservative hypersurfaces in pseudo-Riemannian space forms and their Gauss map

We first present a survey about recent results on biconservative hypersurfaces in the Minkowski space E4 1, pseudo-Euclidean space E5 2 and Rieamnnian space-form H4. Then we obtain some geometrical properties of these hypersurface families concerning their mean curvature and Gauss map.