Dual Unit Sphere Research Articles

On dual Kadec norms

Let (X,‖⋅‖) be a Banach space such that all w⁎-convergent sequences in the dual unit sphere SX⁎ are also norm convergent. Then the weak⁎ and norm topologies agree on SX⁎. By known results it implies that X has a renorming whose dual is locally uniformly rotund, hence also C1-Fréchet smooth. In particular, X is an Asplund space. Our results also lend an alternative proof of the celebrated Josefson-Nissenzweig theorem.

Dual magnetic curves and flux ruled surfaces

In this paper, we introduce the dual magnetic trajectories traced by the dual Frenet vectors of any dual curves in dual space via the dual Lorentz force, and then we give the characterizations of dual magnetic curves. Moreover, we define the dual flux ruled surfaces corresponding to the Killing vectors on the dual unit sphere and identify the conditions for these surfaces to be minimal or developable. Additionally, we present certain characterizations related to these surfaces. We demonstrate the visual representations of some dual flux ruled surfaces in some examples.

Kinematic Geometry of a Timelike Line Trajectory in Hyperbolic Locomotions

This study utilizes the axodes invariants to derive novel hyperbolic proofs of the Euler–Savary and Disteli formulae. The inflection circle, which is widely recognized, is situated on the hyperbolic dual unit sphere, in accordance with the principles of the kinematic theory of spherical locomotions. Subsequently, a timelike line congruence is defined and its spatial equivalence is thoroughly studied. The formulated assertions degenerate into a quadratic form, which facilitates a comprehensive understanding of the geometric features of the inflection line congruence.

Timelike-Ruled and Developable Surfaces in Minkowski 3-Space E13

In this study, the timelike-ruled and developable surfaces are constructed in Minkowski 3-spaceE13. Using the E. Study map, we demonstrate that dual forms of timelike-ruled and developable surfaces can be obtained from the coordinates and first derivatives of the base curve at the dual hyperbolic unit sphere. This is proposed as a novel method for obtaining timelike-ruled and developable surfaces. Some examples have also been provided.

The Dual Expression of Parallel Equidistant Ruled Surfaces in Euclidean 3-Space

In this study, we examine the dual expression of Valeontis’ concept of parallel p-equidistant ruled surfaces well known in Euclidean 3-space, according to the Study mapping. Furthermore, we show that the dual part of the dual angle on the unit dual sphere corresponds to the p-distance. We call these ruled surfaces we obtained “dual parallel equidistant ruled surfaces” and we briefly denote them with “DPERS”. Furthermore, we find the Blaschke vectors, the Blaschke invariants and the striction curves of these DPERS and we give the relationships between these elements. Moreover, we show the relationships between the Darboux screws, the instantaneous screw axes, the instantaneous dual Pfaff vectors and dual Steiner rotation vectors of these surfaces. Finally, we give an example, which we reinforce this article, and we explain all of these features with the figures on the example. Furthermore, we see that the corresponding dual curves on the dual unit sphere to these DPERS are such that one of them is symmetric with respect to the imaginary symmetry axis of the other.

The forward kinematics of dual rolling contact of one dual unit sphere on another with dual Darboux frame based equation

In this paper, we investigate the forward kinematics of dual rolling contact motion without sliding of one dual unit sphere $\tilde{S}_{m}^{2}$ on the fixed sphere $\tilde{S}_{f}^{2}$ along their dual spherical curves, which correspond to ruled surfaces generated by the straight lines in the real line space $\mathbb{E}^{3}$. We adopt a dual Darboux frame method to develop instantaneous kinematics of dual rolling motion. We obtain some new kinematic equations of rolling motion of the moving sphere $\tilde{S}_{m}^{2}$ with regards to dual unit vectors, dual rolling velocity, and dual geometric invariants. Namely, the dual translational velocity equation of an arbitrary dual point and the equation of the dual angular velocity on the moving sphere $\tilde{S}_{m}^{2}$ are derived. The equation represented by geometric invariants can be handily generalized to suit arbitrary dual spherical curve on $\tilde{S}_{m}^{2}$ and can be differentiated to any order.

Geometric Algebra of Singular Ruled Surfaces

Singular ruled surface is an interesting research object and is the breakthrough point of exploring new problems. However, because of singularity, it’s difficult to study the properties of singular ruled surfaces. In this paper, we combine singularity theory and Clifford algebra to study singular ruled surfaces. We take advantage of the dual number of Clifford algebra to make the singular ruled surfaces transform into the dual singular curves on the dual unit sphere. By using the research method on the singular curves, we give the definition of the dual evolute of the dual front in the dual unit sphere, we further provide the k-th dual evolute of the dual front. Moreover, we consider the ruled surface corresponding to the dual evolute and k-th dual evolute and provide the developable conditions of these ruled surfaces.

A topological characterization of dual strict convexity in Asplund spaces

Let X be an Asplund space. We show that the existence of an equivalent norm on X having a strictly convex dual norm is equivalent to the dual unit sphere SX⁎ (equivalently X⁎) possessing a non-linear topological property called (⁎), which was introduced by J. Orihuela, S. Troyanski and the author.

A new approach to design the ruled surface

This paper considers a kind of design of a ruled surface. The design interconnects some concepts from the fields of computer-aided geometric design (CAGD) and kinematics. Dual unit spherical Bézier-like curves on the dual unit sphere (DUS) are obtained by a novel method with respect to the control points. A dual unit spherical Bézier-like curve corresponds to a ruled surface by using Study’s transference principle and closed ruled surfaces are determined via control points and also, integral invariants of these surfaces are investigated. Finally, the results are illustrated by several examples and the motion interpolation was shown as an embodiment of this method.

The dual spatial quaternionic expression of ruled surfaces

In this paper, the ruled surface which corresponds to a curve on dual unit sphere is rederived with the help of dual spatial quaternions. We extend the term of dual expression of ruled surface using dual spatial quaternionic method. The correspondences in dual space of closed ruled surfaces are quaternionically expressed. As a consequence, the integral invariants of these surfaces and the relationships between these invariants are shown

On the Line Congruences

The purpose of this paper is to find the quantities and surfaces of a line congruence via examining it in the dual space and to represent the results more appropriately for computational approximations. For this, we take mainly two-dual parameter motion on the dual unit sphere (DUS) so, we get a line congruence corresponding this motion by a new method. Thus, the equations of the developable surfaces, the principal surfaces, the focal surfaces and the center surface of the line congruence are found by coordinate functions. The results are illustrated by examples.

Rodrigues parameters on dual hyperbolic unit sphere

Rodrigues parameters depend on the tangent of the half rotation angle in Euclidean space but in Dual space, dual Rodrigues parameters contain both rotation angle and distance corresponding the shortest distance between the straight lines in ℝ3. In this paper, we give Cayley's formula for the dual hyperbolic spherical motion and explain 3×3 type L-Dual skew symmetric matrices by using properties of this formula. Then, we obtain Rodrigues pa­rameters of dual Hyperbolic unit sphere and show that Rodrigues parameters contain the hyperbolic rotation angle which is being between timelike lines and distance which is the minimal Lorentzian distance between the timelike lines of .

Characterizations for Slant Ruled Surfaces in Dual Space

In this paper, we study slant ruled surfaces in dual space by considering E. Study’s mapping. We consider ruled surfaces as spherical curves lying on dual unit sphere and study the notion of “slant ruled surface” by means of dual Darboux frame and obtain some dual characterizations for which the real parts of them coincide with results given by Onder (Slant ruled surfaces in Euclidean 3-space E 3, arXiv:1311.0627v1 [math.DG], 2013).

Evolutes of Hyperbolic Dual Spherical Curve in Dual Lorentzian 3-Space

Based on the E. Study's map, we study a timelike ruled surface as a curve on the hyperbolic dual unit sphere in dual Lorentzian 3-space $\mathbb{D}_{1}^{3}$. Then, as applications of the singularity theory of smooth functions, we define the notation of evolutes for timelike ruled surfaces and establish the relationships between their geometric invariants. Finally, an example of application is introduced and explained in detail.

Evolutes of dual spherical curves for ruled surfaces

E. Study found that there is a one‐to‐one correspondence between the oriented lines in Euclidean three space and the dual points of the dual unit sphere in dual three space, and it has wide applications in Engineering. In this paper, we investigate a ruled surface as a curve on the dual unit sphere by using E. Study's theory. Then we define the notion of evolutes of dual spherical curves for ruled surfaces and establish the relationships between singularities of these subjects and geometric invariants of dual spherical curves. Finally, we give an example to illustrate our findings. Copyright © 2015 John Wiley & Sons, Ltd.

Curvature Motion on Dual Hyperbolic Unit Sphere H<sup>2</sup><sub style="margin-left:-8px;">0</sub>

In this paper, we define dual curvature motion on the dual hyperbolic unit sphere H20 in the dual Lorentzian space D31 with dual signature (+,+-) . We carry the obtained results to the Lorentzian line space R31 by means of Study mapping. Then we make an analysis of the orbits during the dual hyperbolic spherical curvature motion. Finally, we find some line congruences, the family of ruled surfaces and ruled surfaces in R31.

Some remarks on stronger versions of the Boundary Problem for Banach spaces

Let X be a real Banach space. A subset B of the dual unit sphere of X is said to be a boundary for X, if every element of X attains its norm on some functional in B. The well-known Boundary Problem originally posed by Godefroy asks whether a bounded subset of X which is compact in the topology of pointwise convergence on B is already weakly compact. This problem was recently solved by Pfitzner in the positive. In this note we collect some stronger versions of the solution to the Boundary Problem, most of which are restricted to special types of Banach spaces. We shall use the results and techniques of Pfitzner, Cascales et al., Moors and others.

Expression of Dual Euler Parameters Using the Dual Rodrigues Parameters and Their Application to the Screw Transformation

Dual numbers and dual vectors are widely used in spatial kinematics [3,5- 15,18]. Plücker line coordinates of a straight line can be represented by a dual unit vector located at the dual unit sphere (DUS). By this way, the trajectory of the screw axis of a rigid body in 3 R (the real three space) corresponds to a dual curve on the DUS. This correspondence is done through Study Mapping [8,9]. Conversely a dual curve on DUS obtained from the rotations of the DUS represents a rigid body motion in 3 R [8]. The dual Euler parameters are used in defining the screw transformation in 3 R [8], but originally in this paper these parameters are constructed from the Rodrigues and the dual Rodrigues parameters [15].

The dual exponential mapping on dual rotations

Dual numbers are introduced by Clifford in the mid-nineteenth century and they are systematically applied to kinematics by Study (1903) and Kotelnikov (1895). Any point on the dual unit sphere (DUS) corresponds to a straight line in QUOTE (real three space), and vice versa. By this way, there is a one to one correspondence between the dual curves on DUS and the one parameter rigid body motions in QUOTE . Using Cayley Mapping (McCarthy, 1990) we get a relation between the dual rotation matrices and the dual skew symmetric matrices. In this paper, this relation is given by the exponential mapping which can be called the dual exponential mapping. Keywords: Kinematics, study mapping, cayley mapping, dual exponential mapping.

On Closed Ruled Surfaces Concerned with Dual Frenet and Bishop Frames

In this paper; the pitch, the angle of pitch and the dual angle of pitch of ruled surface in ℝ3 which is corresponded to a closed curve on dual unit sphere are investigated. This closed curve is formed by the vectors of Frenet and Bishop frame of a closed curve on dual unit sphere. Also the relation between dual angle of pitch and pitch, angle of pitch is obtained.