Relevant bibliographies by topics / Quaternions space

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Quaternions space'

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Published: 9 November 2024

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Journal articles on the topic "Quaternions space"

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Miškinis, P. "ON THE POSSIBLE EXISTENCE OF NEW FERMIONIC DEGREES OF FREEDOM IN D = 6." Mathematical Modelling and Analysis 8, no. 2 (June 30, 2003): 155–64. http://dx.doi.org/10.3846/13926292.2003.9637220.

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The bispinors formed by quaternion in D = 6 dimensional space‐time are proposed to be treated as new fermionic fields. The gauge nonabelian field is formulated by the quaternions. A new kind of physical object, an extended relativistic quaternionic membrane in D = 6, is discussed.
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Ge, Q. J. "On the Matrix Realization of the Theory of Biquaternions." Journal of Mechanical Design 120, no. 3 (September 1, 1998): 404–7. http://dx.doi.org/10.1115/1.2829166.

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This paper describes a matrix algebra realization of Clifford’s theory of biquaternions. By examining 4 × 4 skew-symmetric matrices, the paper shows the connection between infinitesimal screws in elliptic three-space and vector quaternions. By studying the matrix exponential of the skew-symmetric matrices, the paper also shows how finite screws in elliptic three-space lead to matrix realization of quaternions. Finally, it is shown that line transformations in elliptic three-space lead to double quaternions and that a dual quaternion is a limiting case of a double quaternion.
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Dzwonkowski, Mariusz, and Roman Rykaczewski. "Quaternion Feistel Cipher with an Infinite Key Space Based on Quaternion Julia Sets." Journal of Telecommunications and Information Technology, no. 4 (December 30, 2015): 15–21. http://dx.doi.org/10.26636/jtit.2015.4.979.

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In this paper Quaternion Feistel Cipher (QFC) with an infinite key space based on quaternion Julia sets is proposed. The basic structure of the algorithm is based on the scheme proposed in 2012 by Sastry and Kumar. The proposed algorithm uses special properties of quaternions to perform rotations of data sequences in 3D space for each of the cipher rounds. It also uses Julia sets to form an infinite key space. The plaintext is divided into two square matrices of equal size and written using Lipschitz quaternions. A modular arithmetic was implemented for operations with quaternions. A computer-based analysis has been carried out and obtained results are shown at the end of this paper.
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Puleko, I. V., O. V. Andreev, O. F. Dubina, V. O. Chumakevych, and A. S. Palamarchuk. "MODEL OF MOTION OF UNMANNED AERIAL VEHICLES BASED ON DUAL QUATERNION ALGEBRA." Проблеми створення, випробування, застосування та експлуатації складних інформаційних систем, no. 23 (December 28, 2022): 52–61. http://dx.doi.org/10.46972/2076-1546.2022.23.04.

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The widespread use of unmanned aerial vehicles during warfare has intensified the problem of their management, especially when they are used in large groups. One of the main tasks is to ensure coordinated movement of the group's aircraft in space. Optimizing the movement of each device of the group in three-dimensional space is expedient to use mathematical models. The movement of any unmanned aerial vehicle can be presented as a combination of translational and rotational movements, and its speed as a combination of translational and rotational velocities. Previously, these movements were modeled separately using a system of differential equations or quaternions. In this article, a mathematical model of rotational and translational movements of an aircraft based on the algebra of dual quaternions is developed. Dual quaternions consisting of eight scalars are a compact representation of rigid transformations in space. Therefore, their properties determine the advantage in the course of motion simulation, as they reduce the amount of calculations. Thus, with the help of one dual quaternion, it is possible to provide both translational and rotational motions at once, and the operation of non-commutative multiplication of dual quaternions is used to simulate the movement. The model assumes that the real part of the dual quaternion determines the orientation of the UAV in space, and the dual part determines its position in three-dimensional space. In order to connect aircraft coordinate systems with the model, expressions for the transition from aircraft orientation angles (roll, yaw, and pitch) to dual quaternion parameters and vice versa are obtained. The functionality of the proposed model was confirmed using the developed software for modeling the coordinated movement of aircraft. The software is adapted for graphical display of a large number of aircraft in web browsers with WebGl support. Keywords: motion modeling; rotational and translational movement; unmanned aerial vehicles; quaternions; dual quaternions; algebra of quaternions.
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KUMAR, AWNIYA, SUNIL KUMAR SINGH, and SHEO KUMAR SINGH. "A Note on Moritoh Transforms." Creative Mathematics and Informatics 33, no. 2 (May 14, 2024): 185–201. http://dx.doi.org/10.37193/cmi.2024.02.05.

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Some fundamental properties of the Moritoh wavelet are discussed in this paper. The Mori- toh transform is approximated for ultra-distributions in generalised Sobolev space. The adjoint formula of the Fourier transform is extended to the Moritoh transform. The convolution for quaternion-valued functions is de- fined for a modified representation of quaternions. Furthermore, the quaternionic Moritoh transform is defined with the help of convolution. The inner product relation and the uncertainty principle are also established for the quaternionic Moritoh wavelet transform.
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Gogberashvili, Merab. "(2 + 1)-Maxwell Equations in Split Quaternions." Physics 4, no. 1 (March 17, 2022): 329–63. http://dx.doi.org/10.3390/physics4010023.

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The properties of spinors and vectors in (2 + 2) space of split quaternions are studied. Quaternionic representation of rotations naturally separates two SO(2,1) subgroups of the full group of symmetry of the norms of split quaternions, SO(2,2). One of them represents symmetries of three-dimensional Minkowski space-time. Then, the second SO(2,1) subgroup, generated by the additional time-like coordinate from the basis of split quaternions, can be viewed as the internal symmetry of the model. It is shown that the analyticity condition, applying to the invariant construction of split quaternions, is equivalent to some system of differential equations for quaternionic spinors and vectors. Assuming that the derivatives by extra time-like coordinate generate triality (supersymmetric) rotations, the analyticity equation is reduced to the exact Dirac–Maxwell system in three-dimensional Minkowski space-time.
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ATASOY, Ali, and Faik BABADA˘G. "A new Approach to Hyper Dual Split Quaternions with Different Polar Representation." General Letters in Mathematics 14, no. 3 (September 2024): 75–82. http://dx.doi.org/10.31559/glm2024.14.3.4.

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Hamilton first introduced quaternions in 1843 as a way to represent rotations in three dimensional space, and since then, they have become the important tool in many fields. One advantage of quaternions over other methods of representing rotations is their ability to avoid the problem of gimbal lock, which can occur when using Euler angles. Quaternions also have a relatively simple algebraic structure and can be efficiently implemented in computer algorithms. In recent years, quaternions have been used in the development of virtual reality systems and computer games, where they are used to represent orientations of objects in three-dimensional space. They have also been applied in robotics, control theory, and signal processing. Overall, quaternions have become the valuable tools in many areas of mathematics and engineering, and their usage continue to expand. Sangwine and Bihan introduced a quaternion polar representation that draws inspiration from the Cayley-Dickson form. In their formulation, they express quaternions using a complex modulus and argument. The Cayley-Dickson construction is a mathematical procedure that extends the concept of complex numbers to higher dimensions, paving the way for the development of quaternions. On the other hand, the complex argument represents the direction or orientation of the quaternion in a manner analogous to the argument of a complex number. This approach provides a concise and insightful way to represent quaternions, offering a geometric interpretation that aligns with the principles of complex analysis.
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Etzel, K. R., and J. M. McCarthy. "Interpolation of Spatial Displacements Using the Clifford Algebra of E4." Journal of Mechanical Design 121, no. 1 (March 1, 1999): 39–44. http://dx.doi.org/10.1115/1.2829427.

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In this paper we show that the Clifford Algebra of four dimensional Euclidean space yields a set of hypercomplex numbers called “double quaternions.” Interpolation formulas developed to generate Bezier-style quaternion curves are shown to be applicable to double quaternions by simply interpolating the components separately. The resulting double quaternion curves are independent of the coordinate frame in which the key frames are specified. Double quaternions represent rotations in E4 which we use to approximate spatial displacements. The result is a spatial motion interpolation methodology that is coordinate frame invariant to a desired degree of accuracy within a bounded region of three dimensional space. Examples demonstrate the application of this theory to computing distances between spatial displacement, determining the mid-point between two displacements, and generating the spatial motion interpolating a set of key frames.
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Weng, Zi-Hua. "Forces in the complex octonion curved space." International Journal of Geometric Methods in Modern Physics 13, no. 06 (June 15, 2016): 1650076. http://dx.doi.org/10.1142/s0219887816500766.

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The paper aims to extend major equations in the electromagnetic and gravitational theories from the flat space into the complex octonion curved space. Maxwell applied simultaneously the quaternion analysis and vector terminology to describe the electromagnetic theory. It inspires subsequent scholars to study the electromagnetic and gravitational theories with the complex quaternions/octonions. Furthermore Einstein was the first to depict the gravitational theory by means of tensor analysis and curved four-space–time. Nowadays some scholars investigate the electromagnetic and gravitational properties making use of the complex quaternion/octonion curved space. From the orthogonality of two complex quaternions, it is possible to define the covariant derivative of the complex quaternion curved space, describing the gravitational properties in the complex quaternion curved space. Further it is possible to define the covariant derivative of the complex octonion curved space by means of the orthogonality of two complex octonions, depicting simultaneously the electromagnetic and gravitational properties in the complex octonion curved space. The result reveals that the connection coefficient and curvature of the complex octonion curved space will exert an influence on the field strength and field source of the electromagnetic and gravitational fields, impacting the linear momentum, angular momentum, torque, energy, and force and so forth.
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Cansu, Gizem, Yusuf Yaylı, and İsmail Gök. "A new quaternion valued frame of curves with an application." Filomat 35, no. 1 (2021): 315–30. http://dx.doi.org/10.2298/fil2101315c.

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The aim of the paper is to obtain a new version of Serret-Frenet formulae for a quaternionic curve in R4 by using the method given by Bharathi and Nagaraj. Then, we define quaternionic helices in H named as quaternionic right and left X-helix with the help of given a unit vector field X. Since the quaternion product is not commutative, the authors ([4], [7]) have used by one-sided multiplication to find a space curve related to a given quaternionic curve in previous studies. Firstly, we obtain new expressions by using the right product and the left product for quaternions. Then, we generalized the construction of Serret-Frenet formulae of quaternionic curves. Finally, as an application, we obtain an example that supports the theory of this paper.
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Dissertations / Theses on the topic "Quaternions space"

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Kassalias, Ioannis. "Attitude determination for the three-axis spacecraft simulator (TASS) by application of particle filtering techniques." Thesis, Monterey, Calif. : Springfield, Va. : Naval Postgraduate School ; Available from National Technical Information Service, 2005. http://library.nps.navy.mil/uhtbin/hyperion/05Jun%5FKassalias.pdf.

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Salgueiro, Filipe Nuno Ricardo. "Nonlinear pose control and estimation for space proximity operations: an approach based on dual quaternions." Diss., Georgia Institute of Technology, 2014. http://hdl.handle.net/1853/53055.

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The term proximity operations has been widely used in recent years to describe a wide range of space missions that require a spacecraft to remain close to another space object. Such missions include, for example, the inspection, health monitoring, surveillance, servicing, and refueling of a space asset by another spacecraft. One of the biggest challenges in autonomous space proximity operations, either cooperative or uncooperative, is the need to autonomously and accurately track time-varying relative position and attitude references, i.e., pose references, with respect to a moving target, in order to avoid on-orbit collisions and achieve the overall mission goals. In addition, if the target spacecraft is uncooperative, the Guidance, Navigation, and Control (GNC) system of the chaser spacecraft must not rely on any help from the target spacecraft. In this case, vision-based sensors, such as cameras, are typically used to measure the relative pose between the spacecraft. Although vision-based sensors have several attractive properties, they introduce new challenges, such as no direct linear and angular velocity measurements, slow update rates, and high measurement noise. This dissertation investigates the problem of autonomously controlling and estimating the pose of a chaser spacecraft with respect to a moving target spacecraft, possibly uncooperative. Since this problem is inherently hard, the standard approach in the literature is to split the attitude-tracking problem from the position-tracking problem. Whereas the attitude-tracking problem is relatively simple, since the rotational motion is independent from the translational motion, the position-tracking problem is more complicated, as the translational motion depends on the rotational motion. Hence, whereas strong theoretical results exist for the attitude problem, the position problem typically requires additional assumptions. An alternative, more general approach to the pose control and estimation problems is to consider the fully coupled 6-DOF motion. However, fewer results exist that directly address this higher dimensional problem. The main contribution of this dissertation is to show that dual quaternions can be used to extend the theoretical results that exist for the attitude motion into analogous results for the combined position and attitude motion. Moreover, this dissertation shows that this can be accomplished by (almost) just replacing quaternions by dual quaternions in the original derivations. This is because dual quaternions are built on and are an extension of classical quaternions. Dual quaternions provide a compact representation of the pose of a frame with respect to another frame. Using this approach, three new results are presented in this dissertation. First, a pose-tracking controller that does not require relative linear and angular velocity measurements is derived with vision-based sensors in mind. Compared to existing literature, the proposed velocity-free pose-tracking controller guarantees that the pose of the chaser spacecraft will converge to the desired pose independently of the initial state, even if the reference motion is not sufficiently exciting. In addition, the convergence region does not depend on the gains of the controller. Second, a Dual Quaternion Multiplicative Extended Kalman Filter (DQ-MEKF) is developed from the highly successful Quaternion MEKF (Q-MEKF) as an alternative way to achieve pose-tracking without velocity measurements. Existing dual quaternion EKFs are additive, not multiplicative, and have two additional states. The DQ-MEKF is experimentally validated and compared with two conventional EKFs on the 5-DOF platform of the Autonomous Spacecraft Testing of Robotic Operations in Space (ASTROS) facility at the School of Aerospace Engineering at Georgia Tech. Finally, the velocity-free pose-tracking controller is compared qualitatively and quantitatively to a pose-tracking controller that uses the velocity estimates produced by the DQ-MEKF through a realistic proximity operations simulation. Third, a pose-tracking controller that does not require the mass and inertia matrix of the chaser satellite is suggested. This inertia-free controller takes into account the gravitational acceleration, the gravity-gradient torque, the perturbing acceleration due to Earth's oblateness, and constant -- but otherwise unknown -- disturbance forces and torques. Sufficient conditions on the reference pose are also given that guarantee the identification of the mass and inertia matrix of the satellite. Compared to the existing literature, this controller has only as many states as unknown elements and it does not require a priori known upper bounds on any states or parameters. Finally, the inertia-free pose-tracking controller and the DQ-MEKF are tested on a high-fidelity simulation of the 5-DOF platform of the ASTROS facility and also experimentally validated on the actual platform. The equations of motion of the 5-DOF platform, on which the high-fidelity simulation is based, are derived for three distinct cases: a 3-DOF case, a 5-DOF case, and a (2+1)-DOF case. Four real-time experiments were run on the platform. In the first, a sinusoidal reference attitude with respect to the inertial frame is tracked using VSCMGs. In the second, a constant reference attitude is maintained with respect to a target object using VSCMGs and measurements from a camera. In the third, the same sinusoidal reference attitude with respect to the inertial frame tracked in the first experiment is now tracked using cold-gas thrusters. Finally, in the fourth and last experiment, a time-varying 5-DOF reference pose with respect to the inertial frame is tracked using cold-gas thrusters.
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Parcollet, Titouan. "Quaternion neural networks A survey of quaternion neural networks - Chapter 2 Real to H-space Autoencoders for Theme Identification in Telephone Conversations - Chapter 7." Thesis, Avignon, 2019. http://www.theses.fr/2019AVIG0233.

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Au cours des dernières années, l’apprentissage profond est devenu l’approche privilégiée pour le développement d’une intelligence artificielle moderne (IA). L’augmentation importante de la puissance de calcul, ainsi que la quantité sans cesse croissante de données disponibles ont fait des réseaux de neurones profonds la solution la plus performante pour la resolution de problèmes complexes. Cependant, la capacité à parfaitement représenter la multidimensionalité des données réelles reste un défi majeur pour les architectures neuronales artificielles.Pour résoudre ce problème, les réseaux de neurones basés sur les algèbres des nombres complexes et hypercomplexes ont été développés. En particulier, les réseaux de neurones de quaternions (QNN) ont été proposés pour traiter les données tridi- mensionnelles et quadridimensionnelles, sur la base des quaternions représentant des rotations dans notre espace tridimensionnel. Malheureusement, et contrairement aux réseaux de neurones à valeurs complexes qui sont de nos jours acceptés comme une alternative aux réseaux de neurones réels, les QNNs souffrent de nombreuses lacunes qui sont en partie comblées par les différents travaux détaillés par ce manuscrit.Ainsi, la thèse se compose de trois parties qui introduisent progressivement les concepts manquants, afin de faire des QNNs une alternative aux réseaux neuronaux à valeurs réelles. La premiere partie présente et répertorie les précédentes découvertes relatives aux quaternions et aux réseaux de neurones de quaternions, afin de définir une base pour la construction des QNNs modernes.La deuxième partie introduit des réseaux neuronaux de quaternions état de l’art, afin de permettre une comparaison dans des contextes identiques avec les architectures modernes traditionnelles. Plus précisément, les QNNs étaient majoritairement limités par leurs architectures trop simples, souvent composées d’une seule couche cachée comportant peu de neurones. Premièrement, les paradigmes fondamentaux, tels que les autoencodeurs et les réseaux de neurones profonds sont présentés. Ensuite, les très répandus et étudiés réseaux de neurones convolutionnels et récurrents sont étendus à l’espace des quaternions. De nombreuses experiences sur différentes applications réelles, telles que la vision par ordinateur, la compréhension du langage parlé ainsi que la reconnaissance automatique de la parole sont menées pour comparer les modèles de quaternions introduits aux réseaux neuronaux conventionnels. Dans ces contextes bien spécifiques, les QNNs ont obtenus de meilleures performances ainsi qu’une réduction importante du nombre de paramètres neuronaux nécessaires à la phase d’apprentissage.Les QNNs sont ensuite étendus à des conditions d’entrainement permettant de traiter toutes les représentations en entrée des modèles de quaternions. Dans un scénario traditionnel impliquant des QNNs, les caractéristiques d’entrée sont manuellement segmentées en quatre composants, afin de correspondre à la representation induite par les quaternions. Malheureusement, il est difficile d’assurer qu’une telle segmentation est optimale pour résoudre le problème considéré. De plus, une segmentation manuelle réduit fondamentalement l’application des QNNs à des tâches naturellement définies dans un espace à au plus quatre dimensions. De ce fait, la troisième partie de cette thèse introduit un modèle supervisé et un modèle non supervisé permettant l’extraction de caractéristiques d’entrée désentrelacées et significatives dans l’espace des quaternions, à partir de n’importe quel type de signal réel uni-dimentionnel, permettant l’utilisation des QNNs indépendamment de la dimensionnalité des vecteurs d’entrée et de la tâche considérée. Les expériences menées sur la reconnaissance de la parole et la classification de documents parlés montrent que les approches proposées sont plus performantes que les représentations traditionnelles de quaternions
In the recent years, deep learning has become the leading approach to modern artificial intelligence (AI). The important improvement in terms of processing time required for learning AI based models alongside with the growing amount of available data made of deep neural networks (DNN) the strongest solution to solve complex real-world problems. However, a major challenge of artificial neural architectures lies on better considering the high-dimensionality of the data.To alleviate this issue, neural networks (NN) based on complex and hypercomplex algebras have been developped. The natural multidimensionality of the data is elegantly embedded within complex and hypercomplex neurons composing the model. In particular, quaternion neural networks (QNN) have been proposed to deal with up to four dimensional features, based on the quaternion representation of rotations and orientations. Unfortunately, and conversely to complex-valued neural networks that are nowadays known as a strong alternative to real-valued neural networks, QNNs suffer from numerous limitations that are carrefuly addressed in the different parts detailled in this thesis.The thesis consists in three parts that gradually introduce the missing concepts of QNNs, to make them a strong alternative to real-valued NNs. The first part introduces and list previous findings on quaternion numbers and quaternion neural networks to define the context and strong basics for building elaborated QNNs.The second part introduces state-of-the-art quaternion neural networks for a fair comparison with real-valued neural architectures. More precisely, QNNs were limited by their simple architectures that were mostly composed of a single and shallow hidden layer. In this part, we propose to bridge the gap between quaternion and real-valued models by presenting different quaternion architectures. First, basic paradigms such as autoencoders and deep fully-connected neural networks are introduced. Then, more elaborated convolutional and recurrent neural networks are extended to the quaternion domain. Experiments to compare QNNs over equivalents NNs have been conducted on real-world tasks across various domains, including computer vision, spoken language understanding and speech recognition. QNNs increase performances while reducing the needed number of neural parameters compared to real-valued neural networks.Then, QNNs are extended to unconventional settings. In a conventional QNN scenario, input features are manually segmented into three or four components, enabling further quaternion processing. Unfortunately, there is no evidence that such manual segmentation is the representation that suits the most to solve the considered task. Morevover, a manual segmentation drastically reduces the field of application of QNNs to four dimensional use-cases. Therefore the third part introduces a supervised and an unsupervised model to extract meaningful and disantengled quaternion input features, from any real-valued input signal, enabling the use of QNNs regardless of the dimensionality of the considered task. Conducted experiments on speech recognition and document classification show that the proposed approaches outperform traditional quaternion features
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Bouzzit, Aziz. "Ellipsométrie acoustique pour le suivi et la caractérisation de matériaux complexes." Electronic Thesis or Diss., CY Cergy Paris Université, 2024. http://www.theses.fr/2024CYUN1304.

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Les matériaux complexes sont aujourd'hui au cœur des enjeux sociétaux majeurs dans la plupart des grands domaines tels que l'énergie, le transport, l'environnement, la conservation/restauration du patrimoine, la santé ou la sécurité. En effet, de par les opportunités d'innovation offertes en matière de fonctionnalités, ces matériaux suscitent de nouvelles problématiques d'analyse et de compréhension multi-physiques et multi-échelles. Il en va de même pour l'instrumentation nécessaire à leur caractérisation.Répandues dans le domaine de la caractérisation non destructive des milieux complexes, les méthodes acoustiques utilisent les propriétés de propagation des ondes mécaniques dans ces matériaux pouvant être hétérogènes et anisotropes.Dans une approche multi-échelle, l'intérêt des méthodes ultrasonores est d'être particulièrement sensibles à leurs propriétés mécaniques, telles que l'élasticité, la rigidité et la viscosité. La nature hétérogène et multiphasique d'un milieu complexe conduit ainsi à la notion de milieu viscoélastique, caractérisé par les coefficients de Lamé généralisés complexes (��∗, ��∗) et leur variation en fonction de la fréquence.L'objectif de cette thèse est de développer une méthode de caractérisation de ces matériaux complexes viscoélastiques qui permette de mesurer simultanément la variation des deux coefficients de Lamé généralisés complexes (��∗, ��∗) en fonction de la fréquence. L'approche proposée est de suivre, dans l'espace et dans le temps, la propagation de l'onde de Rayleigh et d'extraire ses paramètres ellipsométriques (ellipticité χ et orientation θ) en complément des paramètres propagatifs (k' et k'') classiquement déterminés. Basée sur la détection de l'onde par vibrométrie laser 3D à la surface du matériau complexe, et au moyen de l'analyse de Gabor 2D dans l'espace des Quaternions, l'estimation de l'ensemble des paramètres - propagatifs et ellipsométriques - donne accès à la caractérisation complète du milieu avec cette seule onde de Rayleigh.Les développements théoriques proposés dans ce travail, ainsi que les résultats expérimentaux et issus de simulation, confirment l'intérêt de l'ellipsométrie acoustique pour la caractérisation de ces matériaux complexes
Complex materials are at the heart of major societal challenges in most major fields such as energy, transport, environment, heritage conservation/restoration, health and safety. Because of the opportunities for innovation offered in terms of features, these materials are giving rise to new problems of multi-physical and multi-scale analysis and understanding. The same applies to the instrumentation needed to characterize them.Acoustic methods, which are widely used in the non-destructive characterization of complex media, make use of the propagation properties of mechanical waves in these materials, which can be heterogeneous and anisotropic.In a multi-scale approach, the advantage of ultrasonic methods is that they are particularly sensitive to mechanical properties such as elasticity, rigidity and viscosity. The heterogeneous and multiphase nature of a complex medium thus leads to the notion of a viscoelastic medium, characterized by generalized complex Lamé coefficients (��∗, ��∗) and their variation as a function of frequency.The objective of this thesis is to develop a method for characterizing these complex viscoelastic materials that simultaneously measures the variation of the two generalized complex Lamé coefficients (��∗, ��∗) versus the frequency. The proposed approach is to follow, in space and in time, the propagation of the Rayleigh wave and to extract its ellipsometric parameters (ellipticity χ and orientation θ) in addition to the propagation parameters (k' and k'') conventionally determined. Based on the wave detection by 3D laser vibrometry at the surface of the complex material, and by means of 2D Gabor analysis in Quaternion space, the estimation of propagation and ellipsometric parameters gives access to the complete characterization of the complex material only by studying the interaction of a Rayleigh wave with the medium.The theoretical developments proposed in this work, together with experimental and simulation results, confirm the value of acoustic ellipsometry for characterizing these complex materials
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Silva, Rênad Ferreira da. "Transformações Geométricas no Plano e no Espaço." Universidade Federal da Paraíba, 2013. http://tede.biblioteca.ufpb.br:8080/handle/tede/7476.

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Abstract: In this work we study some geometric transformations in the plane and the space. Initially, we present some special types of transformations in the plane and find the matrix of each of these transformations. In the second part we discourse the transformations in the space, emphasizing the rotations. We will use the angles of Euler to determine a rotation in the space around the Cartesian axes and define an equation which allows to rotate a vector around any axis. We also discuss the homogeneous spaces aiming the matrix representation of transformations of translation. Finally, we use the structure of the quaternions group to present a second form to rotation vectors and composition of rotations in the space. We emphasize that this study is essential to describe the motion of objects in the plane and in the space.
Neste trabalho estudamos algumas das transformações geométricas no Plano e no Espaço. Inicialmente, apresentamos alguns tipos de transformações especiais no Plano e encontramos a matriz de cada uma destas transformações. Na segunda parte abordamos as transformações no Espaço, dando ênfase as rotações. Utilizamos os ângulos de Euler para determinar uma rotação no espaço em torno dos eixos cartesianos e definimos uma equação que permite rotacionar um vetores em torno de um eixo qualquer. Também abordamos os espaços homogêneos objetivando a representa ção matricial da transformação de translação. Por último, usamos a estrutura do grupo dos Quatérnios para apresentar uma segunda forma de fazer rotações de vetores e composição de rotações no espaço. Ressaltamos que este estudo é fundamental para descrever o movimento de objetos no plano e no espaço.
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Mostovoy, J. "Symmetric products and quaternion cycle spaces." Thesis, University of Edinburgh, 1997. http://hdl.handle.net/1842/11203.

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The objects of study in this thesis are symmetric products and spaces of algebraic cycles. The first new result concerns symmetric products and it describes the geometry of truncated symmetric products (or, in other terminology, symmetric products modulo 2). We prove that if M is a closed compact connected triangulable manifold, a necessary and sufficient condition for its symmetric products modulo 2 to be manifolds is that M is a circle. We also show that the symmetric products of the circle modulo 2 are homeomorphic to real projective spaces and give an interpretation of this homeomorphism as a real topological analogue of Vieta's theorem. The second result concerns the spaces of real algebraic cycles, first studied by T.K. Lam. We describe a method of calculating the homotopy groups of the spaces of real cycles with integral coefficients on projective spaces; we give an explicit formula for the groups which lie in the "stable range". The third result (or, rather, a group of results) is the construction of a quaternionic analogue of Lawson's theory of algebraic cycles. We define quaternionic objects as those, which are invariant (in the case of varieties) or equivalent (in the case of polynomials) with respect to a free involution on CP2n+1, induced by the action of the quaternion j on Hn. Basic properties of quaternionic algebraic cycles are studied; a rational "quaternionic suspension theorem" is proved and the spaces of quaternionic cycles with rational coefficients on CP2n+1 are described. We also present a method of calculating the Betti numbers of the spaces of quaternionic cycles of degree 2 and odd codimension on CP∞. Some other results that are included in the thesis are a twisted version of the Dold-Thom theorem and an interpretation of the Kuiper-Massey theorem via symmetric products. After the main results on quaternionic cycles were proved, the author learned that similar results were obtained by Lawson, Lima-Filho and Michelson. Their version of the quaternionic suspension theorem is stronger and requires more sophisticated machinery for the proof.
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Voelkel, Konrad [Verfasser], and Matthias [Akademischer Betreuer] Wendt. "Motivic cell structures for projective spaces over split quaternions." Freiburg : Universität, 2016. http://d-nb.info/1122831854/34.

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Boote, Yumi. "On the symmetric square of quaternionic projective space." Thesis, University of Manchester, 2016. https://www.research.manchester.ac.uk/portal/en/theses/on-the-symmetric-square-of-quaternionic-projective-space(9ac64fc3-60b7-449e-8f5a-264a62b1429b).html.

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The main purpose of this thesis is to calculate the integral cohomology ring of the symmetric square of quaternionic projective space, which has been an open problem since computations with symmetric squares were first proposed in the 1930's. The geometry of this particular case forms an essential part of the thesis, and unexpected results concerning two universal Pin(4) bundles are also included. The cohomological computations involve a commutative ladder of long exact sequences, which arise by decomposing the symmetric square and the corresponding Borel space in compatible ways. The geometry and the cohomology of the configuration space of unordered pairs of distinct points in quaternionic projective space, and of the Thom space MPin(4), also feature, and seem to be of independent interest.
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Scott, Richard A. (Richard Allan). "Real, complex and quaternionic toric spaces." Thesis, Massachusetts Institute of Technology, 1993. http://hdl.handle.net/1721.1/46317.

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Granja, Gustavo 1971. "Self maps of quaternionic projective spaces." Thesis, Massachusetts Institute of Technology, 1997. http://hdl.handle.net/1721.1/42690.

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Books on the topic "Quaternions space"

1

Edmonds, James D. Relativistic reality: A modern view. Singapore: World Scientific, 1997.

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Krieg, Aloys. Modular Forms on Half-Spaces of Quaternions. Berlin, Heidelberg: Springer Berlin Heidelberg, 1985. http://dx.doi.org/10.1007/bfb0075946.

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Alpay, Daniel, Fabrizio Colombo, and Irene Sabadini. Quaternionic de Branges Spaces and Characteristic Operator Function. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-38312-1.

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Edmonds, J. D. Relativistic Reality: A Modern View. World Scientific Publishing Co Pte Ltd, 1997.

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Edmonds, J. D. Relativistic Reality: A Modern View. World Scientific Publishing Co Pte Ltd, 1997.

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Relativistic Reality: A Modern View (Knots and Everything, Vol 12). World Scientific Publishing Company, 1998.

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Lambek, Joachim. Six-Dimensional Lorentz Category. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198748991.003.0014.

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Joachim Lambek had a longstanding interest in the use of quaternions as a tool for explaining fundamental aspects of special relativity, dating from his days as a doctoral student to the end of his career. It is known (since the beginning of the twentieth century) that many areas of theoretical physics may be represented by quaternions with complex coefficients (so called “biquaternions”). This posthumous chapter illustrates how time may (or even should) be represented by three dimensions, so that space–time is represented by a six-dimensional Lorentz category (three space coordinates and three temporal coordinates).
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Spacecraft Modeling, Attitude Determination, and Control: Quaternion-Based Approach. Taylor & Francis Group, 2019.

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Yang, Yaguang. Spacecraft Modeling, Attitude Determination, and Control: Quaternion-Based Approach. Taylor & Francis Group, 2019.

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Yang, Yaguang. Spacecraft Modeling, Attitude Determination, and Control: Quaternion-Based Approach. Taylor & Francis Group, 2019.

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Book chapters on the topic "Quaternions space"

1

Vince, John. "Quaternions in Space." In Quaternions for Computer Graphics, 89–129. London: Springer London, 2011. http://dx.doi.org/10.1007/978-0-85729-760-0_7.

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Vince, John. "Quaternions in Space." In Quaternions for Computer Graphics, 129–75. London: Springer London, 2021. http://dx.doi.org/10.1007/978-1-4471-7509-4_8.

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Vince, John. "Quaternions in Space." In Mathematics for Computer Graphics, 261–83. London: Springer London, 2022. http://dx.doi.org/10.1007/978-1-4471-7520-9_12.

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Vince, John. "Quaternions in Space." In Mathematics for Computer Graphics, 261–83. London: Springer London, 2022. http://dx.doi.org/10.1007/978-1-4471-7520-9_12.

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Vince, John. "Quaternion Transforms in Space." In Rotation Transforms for Computer Graphics, 155–80. London: Springer London, 2011. http://dx.doi.org/10.1007/978-0-85729-154-7_11.

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Xiao, Tingting, and Wanshe Li. "A Novel Robust Adaptive Color Image Watermarking Scheme Based on Artificial Bee Colony." In Proceeding of 2021 International Conference on Wireless Communications, Networking and Applications, 1006–17. Singapore: Springer Nature Singapore, 2022. http://dx.doi.org/10.1007/978-981-19-2456-9_101.

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AbstractThis paper proposes a new robust adaptive watermarking scheme based on dual-tree quaternion wavelet and artificial bee colony, wherein the host images and watermark images are both color images. Color host images and watermark images in RGB space are transformed into YCbCr space. Then, apply Arnold chaotic map on their luminance components and use the artificial bee colony optimization algorithm to generate embedding watermark strength factor. Dual-tree quaternion wavelet transform is performed on the luminance component of the scrambled host image. Apply singular value decomposition on its low-frequency amplitude sub-band to obtain the principal component (PC). Embed the watermark into the principal component. Analysis and experimental results show that the proposed scheme is better as compared to the RDWT-SVD scheme and the QWT-DCT scheme.
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Cecil, Thomas E., and Patrick J. Ryan. "Hypersurfaces in Quaternionic Space Forms." In Springer Monographs in Mathematics, 533–51. New York, NY: Springer New York, 2015. http://dx.doi.org/10.1007/978-1-4939-3246-7_9.

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Hitchin, Nigel. "Quaternionic Kähler Moduli Spaces." In Riemannian Topology and Geometric Structures on Manifolds, 49–61. Boston, MA: Birkhäuser Boston, 2009. http://dx.doi.org/10.1007/978-0-8176-4743-8_3.

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Colombo, Fabrizio, Jonathan Gantner, and David P. Kimsey. "Quaternionic Operators on a Hilbert Space." In Spectral Theory on the S-Spectrum for Quaternionic Operators, 187–217. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-03074-2_9.

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Yau, Donald. "Maps to Spaces in the Genus of Infinite Quaternionic Projective Space." In Categorical Decomposition Techniques in Algebraic Topology, 293–302. Basel: Birkhäuser Basel, 2003. http://dx.doi.org/10.1007/978-3-0348-7863-0_16.

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Conference papers on the topic "Quaternions space"

1

Zhou, Yizhi, Yufan Liu, and Xuan Wang. "Distributed Estimation for a 3-D Moving Target in Quaternion Space with Unknown Correlation." In 2024 IEEE Conference on Control Technology and Applications (CCTA), 394–99. IEEE, 2024. http://dx.doi.org/10.1109/ccta60707.2024.10666607.

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LEMAÎTRE, GEORGES, and RICHARD L. AMOROSO. "Quaternions and Elliptical Space: (Quaternions et Espace Elliptique)." In Unified Field Mechanics II: Preliminary Formulations and Empirical Tests, 10th International Symposium Honouring Mathematical Physicist Jean-Pierre Vigier. WORLD SCIENTIFIC, 2017. http://dx.doi.org/10.1142/9789813232044_0008.

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Purwar, Anurag, and Q. J. Ge. "Polar Decomposition of Unit Dual Quaternions." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-70882.

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This paper seeks to extend the notion of polar decomposition from matrix algebra to dual quaternion algebra. The goal is to obtain a simple, efficient and explicit method for determining the polar decompositions (PD) of spatial displacements in Euclidean three-space that belong to a special Euclidean Group known as SE(3). It has been known that such a decomposition is equivalent to the projection of an element of SE(3) onto SO(4) that yields hyper spherical displacements that best approximate rigid-body displacements. It is shown in this paper that a dual quaternion representing an element of SE(3) can be decomposed into a pair of unit quaternions, called double quaternion, that represents an element of SO(4). Furthermore, this decomposition process may be interpreted as the projection of a point in four-dimensional space onto a unit hypersphere. Examples are provided to illustrate that the results obtained from this dual-quaternion based polar decomposition are same as those obtained from the matrix based polar decomposition.
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Etzel, Karl R., and J. Michael McCarthy. "Spatial Motion Interpolation in an Image Space of SO(4)." In ASME 1996 Design Engineering Technical Conferences and Computers in Engineering Conference. American Society of Mechanical Engineers, 1996. http://dx.doi.org/10.1115/96-detc/mech-1164.

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Abstract Many applications in robotics, spatial mechanism design, and mechanical systems animation require the specification of the movement of a rigid body through space. In this paper, we apply Shoemake’s quaternion interpolation formula to pairs of quaternions, known as biquaternions, to obtain an interpolation procedure for spatial motion. A biquaternion represents a rotation in four dimensional space, an element of SO(4). Because a spatial displacement can be viewed as special case of a four dimensional rotation, we can use a biquaternion as an image point of a spatial configuration and a biquaternion curve to represent spatial movement. We present an example of biquaternion interpolation using four key configurations. The result is a smooth movement in SO(4), which we then map to a spatial movement. A comparison with a similar technique based on dual quaternions is provided.
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Ge, Qiaode Jeffrey, Zihan Yu, Mona Arbab, and Mark Langer. "On the Computation of the Average of Spatial Displacements." In ASME 2022 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2022. http://dx.doi.org/10.1115/detc2022-90156.

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Abstract Many applications in biomechanics and medical imaging call for the analysis of the kinematic errors in a group of patients statistically using the average displacement and the standard deviations from the average. This paper studies the problem of computing the average displacement from a set of given spatial displacements using three types of parametric representations: Euler angles and translation vectors, unit quaternions and translation vectors, and dual quaternions. It has been shown that the use of Euclidean norm in the space of unit quaternions reduces the problem to that of computing the average for each quaternion component separately and independently. While the resulting algorithm is simple, the change of the sign of a unit quaternion could lead to an incorrect result. A novel kinematic measure based on dual quaternions is introduced to capture the separation between two spatial displacement. This kinematic measure is then used to formulate a constrained least squares minimization problem. It has been shown that the problem decomposes into that of finding the optimal translation vector and the optimal unit quaternion. The former is simply the centroid of the set of given translation vectors and the latter can be obtained as the eigenvector corresponding to the least eigenvalue of a 4 × 4 positive definite symmetric matrix. It is found that the weight factor used in combining rotations and translations in the formulation does not play a role in the final outcome. Examples are provided to show the comparisons of these methods.
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Ge, Q. J., Jun Wu, Anurag Purwar, and Feng Gao. "Kinematic Convexity of Planar Displacements Based on an Approximately Bi-Invariant Metric." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-87812.

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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.
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7

Valverde, Alfredo, and Panagiotis Tsiotras. "Relative Pose Stabilization using Backstepping Control with Dual Quaternions." In 2018 Space Flight Mechanics Meeting. Reston, Virginia: American Institute of Aeronautics and Astronautics, 2018. http://dx.doi.org/10.2514/6.2018-1980.

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Schutte, Aaron D., and Firdaus E. Udwadia. "Explicit Nonlinear Rotational Controllers Using the Fundamental Equation." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86795.

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In this paper, we present two explicitly generated nonlinear controllers for rest-to-rest rigid body rotational maneuvers in terms of quaternions. The controllers are brought about by applying the fundamental equation of constrained motion to both the rotational dynamics and rotational control of rigid bodies. The first controller yields asymptotic stability at a desired orientation while allowing the stabilization to occur exactly along a pre-selected trajectory for three of the four components that make-up the quaternion. The second controller provides global stability at the desired orientation allowing stable motion to occur from any point in quaternion space. Numerical examples are provided showing the qualitative behavior that both rotational controllers yield when applied to a rigid body.
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9

Purwar, Anurag, Zhe Jin, and Q. J. Ge. "Rational Motion Interpolation Under Kinematic Constraints of Spherical 6R Closed Chains." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-35727.

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The work reported in this paper brings together the kinematics of spherical closed chains and the recently developed freeform rational motions to study the problem of synthesizing rational interpolating motions under the kinematic constraints of spherical 6R closed chains. The results presented in this paper are extension of our previous work on the synthesis of piecewise rational spherical motions for spherical open chains. The kinematic constraints under consideration are workspace related constraints that limit the position of the links of spherical closed chains in the Cartesian space. Quaternions are used to represent spherical displacements. The problem of synthesizing smooth piecewise rational motions is converted into that of designing smooth piecewise rational curves in the space of quaternions. The kinematic constraints are transformed into geometric constraints for the design of quaternion curves. An iterative algorithm for constrained motion interpolation is presented that detects the violation of the kinematic constraints by searching for those extreme points of the quaternion curve that do not satisfy the constraints. Such extreme points are modified so that the constraints are satisfied and the resulting new points are added to the ordered set of the initial positions to be interpolated. An example is presented to show how this algorithm produces smooth spherical rational spline motions that satisfy the kinematic constraints of a spherical 6R closed chain. The algorithm can also be used for the synthesis of rational interpolating motions that approximate the kinematic constraints of spherical 5R and 4R closed chains within a user-defined tolerance.
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10

Zu, Yue, Unsik Lee, and Ran Dai. "Distributed Motion Estimation of Space Objects Using Dual Quaternions." In AIAA/AAS Astrodynamics Specialist Conference. Reston, Virginia: American Institute of Aeronautics and Astronautics, 2014. http://dx.doi.org/10.2514/6.2014-4296.

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