Twisted Surfaces with Null Rotation Axis in Minkowski 3-Space

We examine curvature properties of twisted surfaces with null rotation axis in Minkowski 3-space. That is, we study surfaces that arise when a planar curve is subject to two synchronized rotations, possibly at different speeds, one in its supporting plane and one of this supporting plane about an axis in the plane. Moreover, at least one of the two rotation axes is a null axis. As is clear from its construction, a twisted surface generalizes the concept of a surface of revolution. We classify flat, constant Gaussian curvature, minimal and constant mean curvature twisted surfaces with a null rotation axis. Aside from pseudospheres, pseudohyperbolic spaces and cones, we encounter B-scrolls in these classifications. The appearance of B-scrolls in these classifications is of course the result of the rotation about a null axis. As for the cones in the classification of flat twisted surfaces, introducing proper coordinates, we prove that they are determined by so-called Clelia curves. With a Clelia curve we mean a curve that has linear dependent spherical coordinates.     

On Developable Ruled Surfaces in Minkowski Space

The aim of this study is to obtain the distribution parameter of a ruled surface generated by a straight line in Frenet trihedron moving along two different spacelike curves with the same parameter. At this time, the Frenet frames of these curves are not the same. We have moved the director vector of the first curve along the second curve. It is shown that the ruled surface is developable if and only if the base curve is helix. In addition, some results and theorems are presented with special cases.     

三维Minkowski空间中类空轴旋转曲面的分类

研究了三维Minkowski空间中满足ΔG=φ(G+C)条件的类空轴旋转曲面,并给出了该类曲面的分类.主要结论为上述条件中当C为零向量时该曲面拥有常值平均曲率;当C为非零向量时,该曲面或是一类圆锥面、或是二类圆锥面.     

On the Curvature of the Indicatrix Surface in Three-Dimensional Minkowski Spaces

Periodica Mathematica Hungarica - As it is well-known a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. If the square of the Minkowski functional...     

Representation Formulas of Curves in a Two- and Three-Dimensional Lightlike Cone

In this paper we give the representation formulas for spacelike curves in two-dimensional lightlike cone ${{\mathbb Q}^2}$ and three-dimensional lightlike cone ${{\mathbb Q}^3}$ . Using these formulas we discuss the properties and structures of cone curves in ${{\mathbb Q}^2}$ and ${{\mathbb Q}^3}$ . Some examples are also given.     

The Expanding Minkowski Space

Under the common relativistic velocity composition law the relativistically admissible velocities form a weakly associative-commutative group (WACG) rather than an ordinary group. A study of the resulting abstract WACG led to the development of a method of constructing other concrete WACGs, two of which are presented here. Due to their similarity to the standard Lorentz transformation group and to its associated weakly associative-commutativegroup of velocities, the presented two WACGs provide a nonstandard Lorentz transformation acting on a nonstandard, expanding Minkowski time-space. The resulting nonstandard Lorentz group, in turn, may be used to test some basic ideas concerning special theory of relativity and its underlying Minkowski space.     

On Conformal Infinity and Compactifications of the Minkowski Space

Using the standard Cayley transform and elementary tools it is reiterated that the conformal compactification of the Minkowski space involves not only the “cone at infinity” but also the 2-sphere that is at the base of this cone. We represent this 2-sphere by two additionally marked points on the Penrose diagram for the compactified Minkowski space. Lacks and omissions in the existing literature are described, Penrose diagrams are derived for both, simple compactification and its double covering space, which is discussed in some detail using both the U(2) approach and the exterior and Clifford algebra methods. Using the Hodge *{star} operator twistors (i.e. vectors of the pseudo-Hermitian space H _2,2) are realized as spinors (i.e., vectors of a faithful irreducible representation of the even Clifford algebra) for the conformal group SO(4, 2)/Z ₂. Killing vector fields corresponding to the left action of U(2) on itself are explicitly calculated. Isotropic cones and corresponding projective quadrics in H _p,q are also discussed. Applications to flat conformal structures, including the normal Cartan connection and conformal development has been discussed in some detail.     

Geometry of Two- and Three-Dimensional Minkowski Spaces

A class of centrally-symmetric convex 12-topes (12-hedrons) in is described such that for an arbitrary prescribed norm on each polyhedron in the class can be inscribed in (circumscribed about) the -ball via an affine transformation, and this can be done with large degree of freedom. Bibliography: 5 titles.     

Hyperbolic Harmonic Function in Minkowski Space

The paper discusses the properties of four-dimensional Minkowski space by using Clifford algebra, then gives the concept of hyperbolic harmonic function by constructing a system (P₄) in the four-dimensional Minkowski space, and obtains several properties and a sufficient and necessary condition for the solvability of the system (P₄) .     

The Geometry of Lightlike Hypersurfaces of the de Sitter Space

It is proved that the geometry of lightlike hypersurfaces of the de Sitter space Sn+11 is directly connected with the geometry of hypersurfaces of the conformal space Cn. This connection is applied for a construction of an invariant normalization and an invariant affine connection of lightlike hypersurfaces as well as for studying singularities of lightlike hypersurfaces.     

Spline Split Quaternion Interpolation in Minkowski Space

Spherical spline quaternion interpolation has been done on sphere in Euclidean space using quaternions. In this paper, we have done the spline split quaternion interpolation on hyperbolic sphere in Minkowski space using split quaternions and metric Lorentz. This interpolation curve is called spherical spline split quaternion interpolation in Minkowski space (MSquad).     

General Helices with Timelike Slope Axis in Minkowski 3-Space

In the present paper, we consider the general helices in Minkowski 3-space to have a constant timelike slope axis. As a result, we show that there exists no pseudo null helix with constant timelike slope axis or spacelike helix with timelike principal normal and constant timelike slope axis. Moreover, we obtain the parametric equations of spacelike helix with spacelike principal normal, timelike helix and null Cartan helix with timelike slope axis. We also give some related examples and their figures.     

Structure of the Space of Diametrically Complete Sets in a Minkowski Space

We study the structure of the space of diametrically complete sets in a finite dimensional normed space. In contrast to the Euclidean case, this space is in general not convex. We show that its starshapedness is equivalent to the completeness of the parallel bodies of complete sets, a property studied in Moreno and Schneider (Isr. J. Math. 2012, doi:10.1007/s11856-012-0003-6), which is generically not satisfied. The space in question is, however, always contractible. Our main result states that in the case of a polyhedral norm, the space of translation classes of diametrically complete sets of given diameter is a finite polytopal complex. The proof makes use of a diagram technique, due to P. McMullen, for the representation of translation classes of polytopes with given normal vectors. The paper concludes with a study of the extreme diametrically complete sets.     

Refinement of Vahlen's Theorem for Minkowski Bases of Three-Dimensional Lattices

In the paper, the analog of Vahlen's theorem for Minkowski bases of three-dimensional lattices is sharpened.     

Space curves of constant breadth in Minkowski 3-space

In this paper, we study spacelike and timelike curves of constant breadth in Minkowski 3-space. We show that in Minkowski 3-space spacelike and timelike curves of constant breadth are normal, helices, and spherical curves in some special cases. Furthermore, we give that the total torsion of a closed spacelike curve of constant breadth is zero while the total torsion of a simple closed timelike curve is equal to ${2\pi n, (n \in Z)}$ .