Academic literature on the topic 'Bi-quaternions'

In this study, we define a type of bi-periodic Fibonacci and Lucas numbers which are called bi-periodic Fibonacci and Lucas Gaussian quaternions. We also give the relationship between negabi-periodic Fibonacci and Lucas Gaussian quaternions and bi-periodic Fibonacci and Lucas Gaussian quaternions. Moreover, we obtain the Binet’s formula, generating function, d’Ocagne’s identity, Catalan’s identity, Cassini’s identity, like-Tagiuri’s identity, Honberger’s identity and some formulas for these new type numbers. Some algebraic proporties of bi-periodic Fibonacci and Lucas Gaussian quaternions which are connected between Gaussian quaternions and bi-periodic Fibonacci and Lucas numbers are investigated.

In this paper, we introduce the bi-periodic Jacobsthal and Jacobsthal-Lucas quaternions. We give the Binet formulas and the generating functions for these quaternions. We obtain some well-known identities such as the Cassini, Catalan and D’ocagne’s identities. Additionally, we give summation formulas and the relationships between bi-periodic Jacobsthal and Jacobsthal-Lucas quaternions.

Abstract We study generalized (doubled) structures in 2D-dimensional Born geometries in which T-duality symmetry is manifestly realized. We show that spacetime structures of Kähler, hyperkähler, bi-hermitian and bi-hypercomplex manifolds are implemented in Born geometries as generalized (doubled) structures. We find that the Born structures and the generalized Kähler (hyperkähler) structures appear as subalgebras of bi-quaternions ℂ × ℍ and split-tetra-quaternions ℍ × Spℍ. We investigate the nature of T-duality for the worldsheet instantons in Born sigma models. This manuscript is based on the original paper [1].

Compact description is given of algebras of poly-numbers: quaternions, bi-quaternions, double (split-complex) and dual numbers. All units of these (and exceptional) algebras are shown to be represented by direct products of 2D vectors of a local basis defined on a fundamental surface. In this math medium a series of equalities identical or similar to known formulas of physical laws is discovered. In particular, a condition of the algebras' stability with respect to transformations of the 2D-basis turns out equivalent to the spinor (Schrödinger–Pauli and Hamilton–Jacobi) equations of mechanics. It is also demonstrated that isomorphism of SO(3, 1) and SO(3, ℂ) groups leads to formulation of a quaternion relativity theory predicting all effects of special relativity but simplifying solutions of relativistic problems in non-inertial frames. Finely it is shown that the Cauchy–Riemann type equations written for functions of quaternion variable repeat vacuum Maxwell equations of electrodynamics, while a quaternion space with non-metricity comprises main relations of Yang–Mills field theory.

Are there solutions of the equation x2 + 1 = 0 ? Carl Fredrich Gauss (1777–1855) conjectured that there was a solution and that it was the square root of - 1 . But since the squares of all real numbers, positive or negative, are positive, Gauss introduced a fanciful idea. His solution to this equation was , which he named i. He integrated i with the real numbers to form a set known as , the complex numbers, where each element in that set was of the form a + bi, where a, . Gauss illustrated this on a graph, the horizontal axis became the real axis and represented the real coefficient, while the vertical axis became the imaginary axis and represented the imaginary coefficient.

La conception assistée par ordinateur (CAO) a été l'un des domaines de recherche fondamental dans la conception automatisée pendant de nombreuses années. Les centres d'intérêt principaux sont les processus exigés pour concevoir différentes pièces, pour simuler leur comportement et pour les inspecter. Le module d’ensemble, qui adresse actuellement le produit entier, par comparaison ; est beaucoup moins étudié et un des processus peu compris dans la conception, sachant que presque tous les logiciels de CAO sont orientés vers les pièces seules. Afin d'offrir un outil pour l'analyse du modèle de produit cette recherche se concentre sur la simulation des opérations d'assemblage et de désassemblage (A/D), une plateforme de simulation étant proposée. Des simulations d'A/D peuvent être effectuées de manière automatisées ou interactives utilisant le matériel informatique standard ou par des simulations immersives en temps réel. Par conséquent, la plateforme proposée est basée sur un nouveau type de processus de simulation adressant jusqu'à deux types de représentations de forme (B-Rep NURBS et polydrique). Cet environnement peut aider les concepteurs pour réaliser une analyse satisfaisante d'ensembles rapidement et peut réduire le temps du développement de produits. Un autre bénéfice de ce travail est la capacité de produire des modèles et des traitements qui améliorent l'intégration des modèles d'assemblages dans des environnements immersifs avec les modèles haptiques et visuels requis. En plus, un opérateur spécifique qui produit des contraintes cinématiques dans le voisinage des configurations de contact est proposé pour déterminer à tout moment les mouvements valides entre les composants. Ainsi, la complexité des algorithmes de détection de collision dans les environnements de simulation en temps réel est réduite. Cette opération est basée sur le modèle mathématique et sur une représentation qui offre la possibilité de décrire toutes les combinaisons valides des phases de montage et de démontage d'un produit
Assembly/Disassembly (A/D) simulations are important to improve design and efficiency of product development processes. In order to get efficient simulation processes it is important to simulate all the possible relative movements between the components in a mechanical assembly. This is important both in the context of interactive simulation and in the context of immersive/real time simulations. If some categories of movements are missing, simulations can loose key configurations, hence they may be no longer meaningful. The scope in research, whiting this thesis deals, in the first time, with a theoretical approach for developing of a kinematical model able to represent all the valid relative movements of a reference component with respect to its surrounding ones, which form a family of trajectories. It is based on the analysis of the three basic movements: translation, rotation and helical ones. In order to determine the compatibility between different families of trajectories, a bi-quaternion is associated to each contact area between the different components. All possible trajectories for each component are analyzed, for the three basic type of movements, in order to find the compatible ones, which leads to the specification of an operator. Thus, the results of all the possible associations are determined and a general combination operator is proposed. The properties of this later are demonstrated as well. The operator can form, in a real time simulation environment, the basis for determining at each moment, the valid movements between components, thus reducing the complexity of collision detection algorithms. The A/D simulations can be performed either from an automated or interactive point of view using standard computer equipment or through immersive and real-time simulation schemes. In order to address this diversity of configurations, a simulation framework was developed. It is based on a new simulation preparation process which allows a simulation process to address up to two types of shape representations, i. E. B-Rep NURBS and polyhedral ones, at the same time, thus handling efficiently the configurations where 3D shape representations of assemblies play a key role. In order to illustrate the simulation process the automatic identification of contacts in a 3D product model and their corresponding list is described. After the identification stage, an interpretation of the results is needed in order to have the complete list with the mechanical contacts for a product. The preparation process is performed within three major stages : model tessellation, surface merging and contacts identification. The framework is based on STEP exchange format. This software environment can assist designers to achieve a satisfactory assembly analysis rapidly and can reduce the lead-time of product development. Further consequences of the present work is its ability to produce models and treatments that improve integration of assembly models in immersive environments taking into account of the haptic and visual models needed. Assembly/Disassembly simulations using haptic devices are facing difficulties while simulating insertion/extraction operations such as removing cylinders from holes for example. In order to address this configuration as well as others, an approach based on the kinematic model and on the simulation framework is proposed

This paper explores the concept of kinematic convexity of planar displacements as an extension of the projective convexity in computational geometry to planar kinematics. This is achieved with the help of planar quaternions which converts planar displacements into points in the space of planar quaternions called the image space. In this way, projective convexity of points in the image space is developed and used as a representation of kinematic convexity of planar displacements. To address the issue of distance metric for planar displacements, we explored the connection between planar quaternions and quaternions and formulated the concept of kinematic convexity in the space of quaternions where a bi-invariant metric exists. An example is provided in the end to illustrate the use of kinematic convexity for estimating the “closest distance” from a fixed body to a moving body undergoing a rational Be´zier motion.

In this paper, we are presenting a framework for capturing human motions using Microsoft Kinect sensor for the purpose of 1) generating task positions for mechanism and robot synthesis, and 2) generation and visualization of B-spline inter-polated and approximated motion from the captured task positions. The theoretical foundation of this work lies in Kinematic Mapping, Dual and Bi-quaternions, and NURBS (Non-Uniform Rational B-spline) geometry. Lately, Kinect has opened doors for creation of natural and intuitive human-machine interactive (HMI) systems in medicine, robotic manipulation, CAD, and many other fields, where visual-sensing and -capture is a central theme. Kinect has made a huge impact in physical therapy area, achieving new benchmarks in tele-rehabilitation by improving physical exercise assessment, monitoring and supervision using the skeletal data. Moreover, Kinect’s depth sensing capability has helped in retrieving depth information required for robotic vision in grasping, object recognition which was previously done using computationally demanding computer vision algorithms. Kinect’s point cloud data with interactive gestures has proven to be useful in various CAD software for conceptual design of shapes. Mechanism synthesis is one of the areas in Kinematics, where Kinect-provided skeletal data can be leveraged to design and develop highly customized end-user collaborated mechanism solutions. We demonstrate that using Kinect, OpenGL, and Openframeworks, we can capture discrete (or, key) rigid body displacements, continuous motions, and generate and visualize rational B-spline motions from captured key positions. Capturing only a few key positions results in significant data savings and also provides a natural way to create tasks for mechanism synthesis problems. The output is a set of dual quaternions and 4 × 4 homogeneous transforms representing a task motion, which can be used as an input for mechanism synthesis applications. The tool produced also allows users to generate trajectories of various points on a moving rigid body interactively. A Kinect-based capture of such motions can help create highly-customized assistive devices for people who suffer from a range of motion-related difficulties due to old age or disabilities.

Geometric Algebra (GA) is popular for its immediate geometric interpretations of algebraic objects and operations. It is based on Clifford Algebra on vector spaces and extends linear algebra of vectors by operations such as an invertible product, i.e. divisions by vectors. This formalism allows for a complete algebra on vectors same as for scalar or complex numbers. It is particularly suitable for rotations in arbitrary dimensions. In Euclidean 3D space quaternions are known to be numerically superior to rotation matrices and already widely used in computer graphics. However, their meaning beyond its numerical formalism often remains mysterious. GA allows for an intuitive interpretation in terms of planes of rotations and extends this concept to arbitrary dimensions by embedding vectors into a higher dimensional, but still intuitively graspable space of multi-vectors. However, out intuition of more than three spatial dimensions is deficient. The space of colors forms a vector space as well, though one of non-spatial nature, but spun by the primary colors red, green, blue. The GA formalism can be applied here as well, amalgamating surprisingly with the notion of vectors and co-vectors known from differential geometry: tangential vectors on a manifold correspond to additive colors red/green/blue, whereas co-vectors from the co-tangential space correspond to subtractive primary colors magenta, yellow, cyan. GA in turn considers vectors, bi-vectors and anti-vectors as part of its generalized multi-vector zoo of algebraic objects. In 3D space vectors, anti-vectors, bi-vectors and covectors are all three-dimensional objects that can be identified with each other, so their distinction is concealed. Confusions arise from notions such as “normal vectors” vs. “axial vectors”. Higher dimensional spaces exhibit the differences more clearly. Using colors instead of spatial dimensions we can expand our intuition by considering "transparency" as an independent, four-dimensional property of a color vector. We can thereby explore 4D GA alternatively to spacetime in special/general relativity. However, even in 4D possibly confusing ambiguities remain between vectors, co-vectors, bi-vectors and bi-co-vectors: bi-vectors and bi-co-vectors - both six-dimensional objects - are visually equivalent. They become unequivocal only in five or higher dimensions. Envisioning five-dimensional geometry is even more challenging to the human mind, but in color space we can add another property, "texture" to constitute a five-dimensional vector space. The properties of a bi-vector and a bi-co-vector becomes evident there: We can still study all possible combinations of colors/transparency/texture visually. This higher-dimensional yet intuitive approach demonstrates the need to distinguish among different kinds of vectors before identifying them in special situations, which also clarifies the meanings of algebraic objects in 3D Euclidean space and allows for better formulations of algorithms in 3D.