The bi-normal fields on spacelike surfaces in ℝ14

2016 ◽  
Vol 09 (03) ◽  
pp. 1650052
Author(s):  
Dang Van Cuong
Keyword(s):  
Ruled Surfaces ◽  
Spacelike Surface ◽  
Surfaces Of Revolution ◽  
Normal Field ◽  
Planar Surfaces ◽  
Spacelike Surfaces ◽  
The Relationship

A normal field [Formula: see text] on a spacelike surface in [Formula: see text] is called bi-normal if [Formula: see text], the determinant of the Weingarten map associated to [Formula: see text], is zero. In this paper, we give a criterion to check if a normal field is bi-normal. Then we study the relationship between the spacelike pseudo-planar surfaces and spacelike pseudo-umbilical surfaces. We also study the bi-normal fields on spacelike ruled surfaces and spacelike surfaces of revolution.

scholarly journals Some Characteristic Properties of Parallelz-Equidistant Ruled Surfaces

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Engin As ◽  
Süleyman Şenyurt
Keyword(s):  
Ruled Surfaces ◽  
Integral Invariants ◽  
Normal Vectors ◽  
The Relationship

Some characteristic properties of two ruled surfaces whose principal normal vectors are parallel along their striction curves inE3are examined by assuming that the distance between two central planes at suitable points is constant,E3. In case of which two ruled surfaces are close, the relationship between the integral invariants of this ruled surfaces is computed.

Geometric invariants and principal configurations on spacelike surfaces immersed in ℝ3,1

2010 ◽  
Vol 140 (6) ◽  
pp. 1141-1160 ◽  
Author(s):  
Pierre Bayard ◽  
Federico Sánchez-Bringas
Keyword(s):  
Second Fundamental Form ◽  
Spacelike Surface ◽  
Geometric Invariants ◽  
Dimensional Minkowski Space ◽  
Numerical Invariants ◽  
Curvature Ellipse ◽  
Curvature Lines ◽  
The Mean ◽  
Asymptotic Lines ◽  
Spacelike Surfaces

We describe the numerical invariants and the curvature ellipse attached to the second fundamental form of a spacelike surface in a four-dimensional Minkowski space. We then study the configuration of the V-principal curvature lines on a spacelike surface when the normal field V is lightlike (the lightcone configuration). We end with some observations on the mean directionally curved lines and on the asymptotic lines on spacelike surfaces.

Isophote curves on spacelike surfaces in Lorentz–Minkowski space

2021 ◽  
pp. 2150180
Author(s):  
Fatih Doğan ◽  
Yusuf Yaylı
Keyword(s):  
Minkowski Space ◽  
Spacelike Surface ◽  
Constant Angle ◽  
Asymptotic Curves ◽  
Lines Of Curvature ◽  
Darboux Frame ◽  
Curves On Surfaces ◽  
Normal Vectors ◽  
Spacelike Surfaces

An isophote curve consists of a locus of surface points whose normal vectors make a constant angle with a fixed vector (the axis). In this paper, we define an isophote curve on a spacelike surface in Lorentz–Minkowski space [Formula: see text] and then find its axis as timelike and spacelike vectors via the Darboux frame. Besides, we give some relations between isophote curves and special curves on surfaces such as geodesic curves, asymptotic curves or lines of curvature.

scholarly journals Four-Dimensional Printing: Design and Fabrication of Smooth Curved Surface Using Controlled Self-Folding

2017 ◽  
Vol 139 (8) ◽  
Author(s):  
Dongping Deng ◽  
Tsz-Ho Kwok ◽  
Yong Chen
Keyword(s):  
Active Material ◽  
Three Dimensional ◽  
Thermal Response ◽  
Curved Surfaces ◽  
4D Printing ◽  
Planar Surfaces ◽  
The Relationship ◽  
Two Sides ◽  
Dimensional Printing ◽  
Printing Method

Traditional origami structures fold along predefined hinges, and the neighboring facets of the hinges are folded to transform planar surfaces into three-dimensional (3D) shapes. In this study, we present a new self-folding design and fabrication approach that has no folding hinges and can build 3D structures with smooth curved surfaces. This four-dimensional (4D) printing method uses a thermal-response control mechanism, where a thermo shrink film is used as the active material and a photocurable material is used as the constraint material for the film. When the structure is heated, the two sides of the film will shrink differently due to the distribution of the constraint material on the film. Consequently, the structure will deform over time to a 3D surface that has no folding hinges. By properly designing the coated constraint patterns, the film can be self-folded into different shapes. The relationship between the constraint patterns and their correspondingly self-folded surfaces has been studied in the paper. Our 4D printing method presents a simple approach to quickly fabricate a 3D shell structure with smooth curved surfaces by fabricating a structure with accordingly designed material distribution.

COMPLETE CMC SPACELIKE SURFACES WITH BOUNDED HYPERBOLIC ANGLE IN GENERALIZED ROBERTSON–WALKER SPACETIMES

2010 ◽  
Vol 07 (06) ◽  
pp. 961-978 ◽  
Author(s):  
MAGDALENA CABALLERO ◽  
ALFONSO ROMERO ◽  
RAFAEL M. RUBIO
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Existence Theorems ◽  
Natural Extension ◽  
Minkowski Spacetime ◽  
Spacelike Surface ◽  
3 Dimensional ◽  
Spacelike Surfaces ◽  
Hyperbolic Angle ◽  
Wide Family

Complete spacelike surfaces with constant mean curvature (CMC) and bounded hyperbolic angle in Generalized Robertson–Walker (GRW) spacetimes, obeying certain natural curvature assumptions, are studied. This boundedness assumption arises as a natural extension of the notion of bounded hyperbolic image of a spacelike surface in the 3-dimensional Lorentz–Minkowski spacetime. The results obtained apply to complete CMC spacelike surfaces lying between two spacelike slices in an GRW spacetime, in the steady state spacetime and in a static GRW spacetime. As an application, uniqueness and non-existence theorems for certain CMC spacelike surface differential equations in a wide family of open GRW spacetimes are given.

scholarly journals Complete maximal spacelike surfaces in an anti-de Sitter space {\bf H}^{4}_{2}(c)

2000 ◽  
Vol 42 (1) ◽  
pp. 139-156
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
De Sitter Space ◽  
Second Fundamental Form ◽  
Spacelike Surface ◽  
De Sitter ◽  
Totally Geodesic ◽  
The Second Fundamental Form ◽  
Spacelike Surfaces ◽  
Hyperbolic Cylinder

In this paper, we prove that if M^2 is a complete maximal spacelike surface of an anti-de Sitter space {\bf H}^{4}_{2}(c) with constant scalar curvature, then S=0, S={-10c\over 11}, S={-4c\over 3} or S=-2c, where S is the squared norm of the second fundamental form of M^{2}. Also(1) S=0 if and only if M^2 is the totally geodesic surface {\bf H}^2(c);(2) S={-4c\over 3} if and only if M^2 is the hyperbolic Veronese surface;(3) S=-2c if and only if M^2 is the hyperbolic cylinder of the totally geodesicsurface {\bf H}^{3}_{1}(c) of {\bf H}^{4}_{2}(c).1991 Mathematics Subject Classifaction 53C40, 53C42.

Spacelike surface geometry

2017 ◽  
Vol 14 (09) ◽  
pp. 1750118
Author(s):  
Süleyman Şenyurt ◽  
Sümeyye Gur
Keyword(s):  
Gauss Curvature ◽  
Intersection Point ◽  
Normal Vector ◽  
Spacelike Surface ◽  
Surface Geometry ◽  
Unit Normal Vector ◽  
Spacelike Curve ◽  
Rotation Vectors ◽  
The Relationship ◽  
Tangent Vectors

In this paper, by considering [Formula: see text] and [Formula: see text] parameter curves on spacelike surface [Formula: see text], [Formula: see text] and [Formula: see text], respectively, and any spacelike curve [Formula: see text] that passes through the intersection point of these parameter curves, we have found the Darboux instantaneous rotation vectors of Darboux trihedrons of these three curves, as follows: [Formula: see text] [Formula: see text] [Formula: see text] and we have obtained the relationship between these vectors as [Formula: see text] where [Formula: see text] and [Formula: see text] are the spacelike angles between tangent vectors of [Formula: see text] and [Formula: see text] curves, and of [Formula: see text] and [Formula: see text] curves, respectively. [Formula: see text] is the unit normal vector of the surface. Besides, we have given Euler, Liouville, Bonnet formulas and Gauss curvature of the spacelike surface with new statement.

scholarly journals Influence of the geometrical researches of surfaces of revolution and translation surfaces on design of unique structures

2019 ◽  
Vol 15 (4) ◽  
pp. 308-314 ◽  
Author(s):  
Gérard Léopold Gbaguidi Aisse
Keyword(s):  
Computer Modeling ◽  
Real Structure ◽  
Building Structures ◽  
Surfaces Of Revolution ◽  
Translation Surfaces ◽  
Fractal Surfaces ◽  
Thin Walled Structures ◽  
The World ◽  
Thin Walled ◽  
The Relationship

Aims of research. The use, design and analysis of architectural and building structures in the form of smooth and composite surfaces have become relevant and in demand lately, which determined the purpose of this article - to analyze the use of analytical surfaces given vector, parametric or explicit equations in real structures. Methods. The relationship between studies on the geometry of surfaces of revolution and transport and the creation of new forms of thin-walled structures and buildings is determined. An example of a real structure is given on each surface. The article does not consider composite, multifaceted, fractal surfaces, as well as surfaces that are not defined analytically. Results. It turned out that only a small number of considered surfaces of these two classes have found application in the world. At the end of the article, a bibliography is presented, which sets out the mathematical side of the design of analytical surfaces, their computer modeling, more detailed information about real structures in the form of the surfaces under consideration.

scholarly journals On parallel p-equidistant ruled surfaces by using mofied orthogonal frame with curvature in E³

2022 ◽  
Vol 40 ◽  
pp. 1-7
Author(s):  
Muhammed T. Sariaydin ◽  
Talat Korpinar ◽  
Vedat Asil
Keyword(s):  
Euclidean Space ◽  
Apex Angle ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Dimensional Euclidean Space ◽  
3 Dimensional ◽  
The Relationship ◽  
Orthogonal Frame

In this paper, it is investigated Ruled surfaces according to modified orthogonal frame with curvature in 3-dimensional Euclidean space. Firstly, we give apex angle, pitch and drall of closed ruled surface in E³. Then,  it characterized the relationship between these invariant of parallel p-equidistant ruled surfaces.

scholarly journals Lorentzian Darboux images of curves on spacelike surfaces in Lorentz–Minkowski 3-space

2016 ◽  
Vol 13 (05) ◽  
pp. 1650066 ◽  
Author(s):  
Noriaki Ito ◽  
Shyuichi Izumiya
Keyword(s):  
Vector Fields ◽  
Moving Frame ◽  
Spacelike Surface ◽  
Regular Curve ◽  
Darboux Frame ◽  
Minkowski Formula ◽  
Spacelike Surfaces

For a regular curve on a spacelike surface in Lorentz–Minkowski [Formula: see text]-space, we have a moving frame along the curve which is called a Lorentzian Darboux frame. We introduce five special vector fields along the curve associated to the Lorentzian Darboux frame and investigate their singularities.

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