Timelike Christoffel pairs in the split-quaternions

We characterize the Christoffel pairs of timelike isothermic surfaces in the four-dimensional split-quaternions. When restricting the receiving space to the three-dimensional imaginary split-quaternions, we establish an equivalent condition for a timelike surface in R(2)(3) to be real or complex isothermic in terms of the existence of integrating factors.

Spin stabilized satellite's attitude propagation with quaternions

This paper presents a study of a modeling scheme for the spin stabilized satellites attitude, entirely developed in terms of quaternion parametrization. The analysis includes numerical propagation of the rotational motion equation, considering the influence of the following torques: aerodynamic, gravity gradient, residual magnetic, eddy currents and the one due to the Lorentz force. Applications are developed considering the Brazilian Spin Stabilized Satellites SCD1 and SCD2, which are quite appropriated for verification and comparison of the theory with the real data generated and processed by the INPE's Satellite Control Center (SCC). The results show that for SCD1 and SCD2 the influence of the eddy current torque is bigger than the others ones, not only due to the orbit altitude, but also to other specific satellites characteristics. The influence of the torque due to Lorentz force is smaller than the others ones because of the dimension and the electrical charges of the SCD1 and SCD2. In all performed tests the errors remained within the dispersion range specified for the attitude determination system of INPE's SCC. The results show the feasibility of using the quaternion attitude parametrization for modeling the satellite dynamics of spin stabilized satellites.

Semi-analytical study of the rotational motion stability of artificial satellites using quaternions

This study at aims performing the stability analysis of the rotational motion to artificial satellites using quaternions to describe the satellite attitude (orientation on the space). In the system of rotational motion equations, which is composed by four kinematic equations of the quaternions and by the three Euler equations in terms of the rotational spin components. The influence of the gravity gradient and the direct solar radiation pressure torques have been considered. Equilibrium points were obtained through numerical simulations using the softwares Matlab and Octave, which are then analyzed by the Routh-Hurwitz Stability Criterion.

A note on the hypercomplex riemann-cauchy like relations for quaternions and laplace equations

In this Note it is worked out a new set of Laplace-Like equations for quaternions through Riemann-Cauchy hypercomplex relations otained earlier [1]. As in the theory of functions of a complex variable, it is expected that this new set of Laplace-Like equations might be applied to a large number of Physical problems, providing new insights in the Classical Fields Theory.

Fourier series for quaternions and the square of the error theorem

In this paper we introduce a type of Hypercomplex Fourier Series based on Quaternions, and discuss on a Hypercomplex version of the Square of the Error Theorem. Since their discovery by Hamilton (Sinegre [1]), quaternions have provided beautifully insights either on the structure of different areas of Mathematics or in the connections of Mathematics with other fields. For instance: I) Pauli spin matrices used in Physics can be easily explained through quaternions analysis (Lan [2]); II) Fundamental theorem of Algebra (Eilenberg [3]), which asserts that the polynomial analysis in quaternions maps into itself the four dimensional sphere of all real quaternions, with the point infinity added, and the degree of this map is n. Motivated on earlier works by two of us on Power Series (Pendeza et al. [4]), and in a recent paper on Liouville’s Theorem (Borges and Mar˜o [5]), we obtain an Hypercomplex version of the Fourier Series, which hopefully can be used for the treatment of hypergeometric partial differential equations such as the dumped harmonic oscillation.

Bulletin of the International association for promoting the study of quaternions and allied systems of mathematics.

Only the numbers for 1910 and 1913 have title Bulletin; the other numbers bear the name in the association only.

Robotics toolbox for MATLAB, release 9 [Software]

The ninth release of the Toolbox, represents over fifteen years of development and a substantial level of maturity. This version captures a large number of changes and extensions generated over the last two years which support my new book “Robotics, Vision & Control”. The Toolbox has always provided many functions that are useful for the study and simulation of classical arm-type robotics, for example such things as kinematics, dynamics, and trajectory generation. The Toolbox is based on a very general method of representing the kinematics and dynamics of serial-link manipulators. These parameters are encapsulated in MATLAB ® objects - robot objects can be created by the user for any serial-link manipulator and a number of examples are provided for well know robots such as the Puma 560 and the Stanford arm amongst others. The Toolbox also provides functions for manipulating and converting between datatypes such as vectors, homogeneous transformations and unit-quaternions which are necessary to represent 3-dimensional position and orientation. This ninth release of the Toolbox has been significantly extended to support mobile robots. For ground robots the Toolbox includes standard path planning algorithms (bug, distance transform, D*, PRM), kinodynamic planning (RRT), localization (EKF, particle filter), map building (EKF) and simultaneous localization and mapping (EKF), and a Simulink model a of non-holonomic vehicle. The Toolbox also including a detailed Simulink model for a quadcopter flying robot.

Concurrent validity of accelerations measured using a tri-axial inertial measurement unit while walking on firm, compliant and uneven surfaces

Although accelerometers are extensively used for assessing gait, limited research has evaluated the concurrent validity of these devices on less predictable walking surfaces or the comparability of different methods used for gravitational acceleration compensation. This study evaluated the concurrent validity of trunk accelerations derived from a tri-axial inertial measurement unit while walking on firm, compliant and uneven surfaces and contrasted two methods used to remove gravitational accelerations: i) subtraction of the best linear fit from the data (detrending), and; ii) use of orientation information (quaternions) from the inertial measurement unit. Twelve older and twelve younger adults walked at their preferred speed along firm, compliant and uneven walkways. Accelerations were evaluated for the thoracic spine (T12) using a tri-axial inertial measurement unit and an eleven-camera Vicon system. The findings demonstrated excellent agreement between accelerations derived from the inertial measurement unit and motion analysis system, including while walking on uneven surfaces that better approximate a real-world setting (all differences <0.16 m.s−2). Detrending produced slightly better agreement between the inertial measurement unit and Vicon system on firm surfaces (delta range: −0.05 to 0.06 vs. 0.00 to 0.14 m.s−2), whereas the quaternion method performed better when walking on compliant and uneven walkways (delta range: −0.16 to −0.02 vs. −0.07 to 0.07 m.s−2). The technique used to compensate for gravitational accelerations requires consideration in future research, particularly when walking on compliant and uneven surfaces. These findings demonstrate trunk accelerations can be accurately measured using a wireless inertial measurement unit and are appropriate for research that evaluates healthy populations in complex environments.

Luonnollisten lukujen joukon määriteltävyys funktioalgebroissa

Let X be a topological space and K the real algebra of the reals, the complex numbers, the quaternions, or the octonions. The functions form X to K form an algebra T(X,K) with pointwise addition and multiplication. We study first-order definability of the constant function set N' corresponding to the set of the naturals in certain subalgebras of T(X,K). In the vocabulary the symbols Constant, +, *, 0', and 1' are used, where Constant denotes the predicate defining the constants, and 0' and 1' denote the constant functions with values 0 and 1 respectively. The most important result is the following. Let X be a topological space, K the real algebra of the reals, the compelex numbers, the quaternions, or the octonions, and R a subalgebra of the algebra of all functions from X to K containing all constants. Then N' is definable in , if at least one of the following conditions is true. (1) The algebra R is a subalgebra of the algebra of all continuous functions containing a piecewise open mapping from X to K. (2) The space X is sigma-compact, and R is a subalgebra of the algebra of all continuous functions containing a function whose range contains a nonempty open set of K. (3) The algebra K is the set of reals or the complex numbers, and R contains a piecewise open mapping from X to K and does not contain an everywhere unbounded function. (4) The algebra R contains a piecewise open mapping from X to the set of the reals and function whose range contains a nonempty open subset of K. Furthermore R does not contain an everywhere unbounded function.

Consistent quaternion interpolation for objective finite element approximation of geometrically exact beam

We explore an isoparametric interpolation of total quaternion for geometrically consistent, strain-objective and path-independent finite element solutions of the geometrically exact beam. This interpolation is a variant of the broader class known as slerp. The equivalence between the proposed interpolation and that of relative rotation is shown without any recourse to local bijection between quaternions and rotations. We show that, for a two-noded beam element, the use of relative rotation is not mandatory for attaining consistency cum objectivity and an appropriate interpolation of total rotation variables is sufficient. The interpolation of total quaternion, which is computationally more efficient than the one based on local rotations, converts nodal rotation vectors to quaternions and interpolates them in a manner consistent with the character of the rotation manifold. This interpolation, unlike the additive interpolation of total rotation, corresponds to a geodesic on the rotation manifold. For beam elements with more than two nodes, however, a consistent extension of the proposed quaternion interpolation is difficult. Alternatively, a quaternion-based procedure involving interpolation of relative rotations is proposed for such higher order elements. We also briefly discuss a strategy for the removal of possible singularity in the interpolation of quaternions, proposed in [I. Romero, The interpolation of rotations and its application to finite element models of geometrically exact rods, Comput. Mech. 34 (2004) 121–133]. The strain-objectivity and path-independence of solutions are justified theoretically and then demonstrated through numerical experiments. This study, being focused only on the interpolation of rotations, uses a standard finite element discretization, as adopted by Simo and Vu-Quoc [J.C. Simo, L. Vu-Quoc, A three-dimensional finite rod model part II: computational aspects, Comput. Methods Appl. Mech. Engrg. 58 (1986) 79–116]. The rotation update is achieved via quaternion multiplication followed by the extraction of the rotation vector. Nodal rotations are stored in terms of rotation vectors and no secondary storages are required.

Legendarius per circulum anni, pars.

Légendier disposé dans l'ordre du calendrier liturgique du 20 juillet au 25 novembre et contenant les saints suivants: ss. Margarita, Maria Magdelene, Jacobus major ap., Stephanus, Laurentius, Hippolytus, assumptio b. Mariae, Bartholomaeus, Augustinus, Johannes Baptista, Aegidius, nativitas b. Mariae, Gorgonius, Protus et Hyacinthus, exaltatio s. Crucis, Euphemia, Matthaeus, Mauritius, Cosmas et Damianus, Michael archangelus, Hieronymus, Leodegarius, Dionysius, Lucas, Simon et Judas, Martinus, Caecilia, Clemens, Catharina. F. 1-130. Legendarius. F. 1-3v. "... passio sancte Margarete virginis"; cf. B.H.L. n° 5306. F. 3v-6v. "... vita sancte Marie Magdalene" [ODO Cluniacensis abbas, sermo 2, excerptum] (P.L. 133, 714B-719C); cf. B.H.L. n° 5440. F. 6v-8v. "... passio sancti Jacobi [majoris] apostoli" incomplet des lignes finales par lacune matérielle, à partir de "decollandi erant dixit [Jacobus...]" [Virtutes apostolorum, de s. Jacobo majore] (Mombritius, 2a ed., II, 37-40 lig. 15); cf. B.H.L. n° 4057; C.A.N.T. n° 272. F. 9-12. "Inventio sancti Stephani prothomartyris" [LUCIANUS presbyter, interprete AVITO presbytero Bracarensi ]. "Domino venerabili Nimpsio [sic] episcopo... [Julianus exponctué et corrigé en] Lucianus... Revelationem que mihi ostensa est...-... aperire dignatus est..." recensio mixta suivie d'un court épilogue: "Bovem appellatum... declarata sunt"; cf. C.P.L. n° 575; B.H.L. Suppl. n° 7851n (E. Vanderlinden, Rev. des études byzantines, IV, 1946, 190-216, version A'; ne fait pas partie des mss. recensés pour l'édition). F. 12-16. "... passio sancti [Sixti et sancti] Laurentii". "In illo tempore Decius Caesar et Valerianus prefectus jusserunt...-... participati sunt omnes"; cf. B.H.L. Suppl. n° 7801 avec var. incipit et n° 4754. F. 16-18. "... passio sancti Ypoliti et sociorum ejus". "Egressus itaque Ypolitus post tercium diem...-... vivere male et regnare cum Christo "; cf. B.H.L. n° 3961 avec var. incipit et explicit. F. 18-31. "... [sermo] beati Ieronimi in assumptione sancte Virginis" [PASCHASIUS RADBERTUS, De Adsumptione b. Mariae] (C.C.C.M., LVI C 109-162; édité aussi parmi les apocryphes de saint Jérôme, P.L., XXX, 126-147); cf. C.P.L. n° 633, ep. 9 (18-29). F. 29-31. Addition contemporaine : "In vigilia assumptionis beate Marie legatur: Secundum Lucam (Lc 11, 27). [M]agne devocionis et fidei...-... ubera que suxisti" [BEDA, In Lc, l. IV, excerptum] divisé en trois paragraphes, correspondant sans doute à trois leçons (P.L., XCII, 479 C-480 B; C.C.S.L., CXX, 236-237 lig. 213-244) (29). — "Sermo iste legatur in nativitate beate Marie virginis"; cf. infra f. 56v. "[A]pprobate consuetudinis est apud christianos...-... pacta cessare" [FULBERTUS CARNOTENSIS, sermo 4] incomplet de la fin qui a été laissée en blanc (P.L. CXLI, 320-324 A; J.M. Canal, dans Rech. théol. anc. méd., XXX (1963), 56-61 lig. 168); ne fait pas partie des mss. répertoriés par J.M. Canal, ibid., XXIX (1962), 36-37; division marginale en neuf, puis en trois fois trois leçons; en marge du titre, une note difficilement lisible identifie l'auteur: "sermo Fulberti ep. Carnotensis" (29v-31). F. 32-35. "... vita [sic pro passio] sancti Bartholomei apostoli" [Virtutes apostolorum, de s. Bartholomaeo]; cf. B.H.L. Suppl. n° 1002a; C.A.N.T. n° 259. F. 35-49v. "[Vita s.] Augustini episcopi", sans prologue [POSSIDIUS]. "Ex provincia affricana civitate...-... perfruar"; cf. B.H.L. n° 785; C.P.L. n° 358 (35-49a lig. 7); suivi de la liste des livres de s. Augustin dans l'ordre des Retractations : "Libros vero quod edidit hic breviter enumerare vel annotare non omissi. Et hoc indicium librorum omnium sancti Augustini. Contra paganos achademicos [sic] libri III. De beata vita liber I. De ordine sacro libri II. De soliloquiis...-... et gratia liber. Requievit autem... V. kal. septembris..." (C.C.S.L., LVII (1984), 1-4, var.); texte proche du ms. lat. 5276, ff. 136v-137, répertorié par A. Wilmart, Miscellanea Agostiniana, Roma, 1931, 157 (Testi e studi, 2) (49a lig. 7-49v). F. 49v-52v. "Inventio capitis sancti Johannis Baptiste"; cf. B.H.L. n° 4296 (49v-51). — "De translatione ejusdem [Angeriacum]"; cf. B.H.L. n° 4297 (51-52v). F. 52v-56v. "... vita sancti Egidii"; cf. B.H.L. n° 93. F. 56v-59v. "De nativitate sancte Marie". "Petis a me petitiunculam opere...-... prefationem habuisse"; cf. B.H.L. Suppl. n° 5345 (56v a-b lig. 27) ; suivi de: "Petitis a me...-... scribi potuerunt. Igitur beata et gloriosissima semper virgo Maria..-... docuerunt Dominum..." (éd. parmi les apocryphes de s. Jérôme, P.L., XXX, 2a ed., 307-317); cf. B.H.L. n° 5344-5343; C.P.L., n° 633, ep. 50 (56v b lig. 27-59v). L'attribution à Paschase Radbert des deux lettres regroupées en une seule faite par C. Lambot, dans Rev. bénéd., XLVI (1934), 271-282, est réfutée par R. Beyers, dans Rev. Théol. et Philos., CXXII (1990), 171-188. Voir sa nouv. éd. dans CC Apocrypha, 10. F. 59v-61v. "Passio sancti Gorgonii [et Dorothei]; cf. B.H.L. n° 3617. F. 61v-62. "[Passio ss.] Prothi et Jacincti"; cf. B.H.L. n° 6977. F. 62-63v. "De exaltatione sancte Crucis". "Tempore illo postquam Constantino Augusto contra Maxentium..."; cf. B.H.L. n° 4178, avec var. incipit. F. 64-68v. "[Passio s.] Eufemie virginis". "Quinto persecutionis anno Diocletiani...-... Completum est autem martyrium... Prisco proconsule Europe..."; cf. B.H.L. n° 2709, avec var. explicit. F. 68v-72v. "[Passio] sancti Mathei apostoli" [Virtutes apostolorum, de s. Jacobo majore]; cf. C.A.N.T. n° 270; B.H.L. n° 5690, avec var. explicit de l'épilogue: "Zaroes autem...-... passio eorum ostendit". F. 72v-76. "[Passio s.] [Marcii corrigé en] Mauricii con [sic] sociis suis" [s. EUCHERIUS LUGDUNENSIS] sans le prologue; cf. B.H.L. n° 5738; C.P.L. n° 490. F. 76-79v. "[Passio ss.] Cosme et Damiani"; cf. B.H.L. Suppl. n° 1975. F. 79v-80. "[In festivitate s.] Michaelis archangeli". "Angelorum quippe et hominum naturam...-... medicina Dei." [GREGORIUS MAGNUS, Hom. in Ev., 34, excerptum]; C.P.L. n° 1711 (P.L., LXXVI, 1249 C-1251 A, §§ 6, fin-9 début) divisé en 8 paragraphes; une interpolation a été ajoutée dans la marge inférieure du f. 79v par une main contemporaine qui a également numéroté les paragraphes en IX leçons, le texte ajouté formant la lectio IIa : "[N]ovem esse angelorum ordines ad Dei judicia...-... principantur."; il s'agit d'un court extrait du sermon Legimus in ecclesiasticis historiis édité par J. E. Cross, dans Traditio, 33 (1977), 108-109 lig. 41-47 (Beda, Homilia subditia 71, P.L., XCIV, 453 C); cf. C.P.P.M., I, 4046. F. 80-82v. "[Vita s.] Jeronimi presbiteri". "Hieronimus noster [corrigé en: presbiter] in oppido Stridonis...-... etatis sue anno in Domino requievit cui..." extraits de la Vie apocryphe de Gennadius (P.L., XXII, 175-184, passim, avec var.); cf. C.P.L. n° 623; B.H.L. n° 3869; Lambert, B.H.M., IIIA, 630 (80-81b lig. 26); suivi du miracle du lion extrait de la Vie du Ps. Sebastianus Casinensis: "Contigit autem hujusmodi miraculum in monasterio... Quadam namque die ingens leo... - asserendo narrantur" (P.L., XXII, 210 lig. 11-213 lig. 11); cf. C.P.L. n° 622; B.H.L. n° 3872 avec var. incipit; Lambert, B.H.M., IIIA, 630 (81b lig. 26-82v). F. 82v-88v. "[Vita s.] Leodegarii". "Igitur sanctus Leodegarius ex progenie...-... postmodum cecum. ... adnecteret opera ibidem" [URSINUS LOGOGIACENSIS] sans le prologue et incomplet de la fin (C.C.S.L., CXVII, 589-632 lig. 14, avec var.); cf. C.P.L. n° 1079a; B.H.L. n° 4851; suivi d'un court extrait omis à sa place plus haut dans le texte: "Deinde vero ire ceperunt... Dei opera ibidem" (ed. cit., 631 § 31 lig. 3-7). F. 88v-98. "[Passio] sancti Dyonisii martyris", texte incomplet par suite de la perte de 2 ff. entre les ff. 96 et 97, le texte s'arrête à "...fideliter adhe[-rebat]" et reprend à "[Domitia-]no per tres Cesares..." (P.L., CVI, 23-40 C et 48 A-50); cf. B.H.L. n° 2175. F. 98-100v. "[Laudatio s.] Luce evvangeliste". "Gloriosus igitur evvangelista Jhesu Christi Lucas natione Syrus...-... ubique confluunt qui ..." [PAULUS DIACONUS, hom. 59] incomplet du prologue (P.L., XCV, 1530-1535, avec var.); cf. B.H.L. n° 4974, d'après ce ms. F. 100v-106. "[Passio ss.] Symonis et Jude apostolorum" avec l'épilogue [ABDIAS, Virtutes Simonis et Judae Thaddaei]; cf. B.H.L. n° 7750-7751; C.A.N.T. n° 284. F. 106-107v. "[Laudatio] sancti Martini archiepiscopi" [ALCUINUS, De vita s. Martini, pars I]; cf. B.H.L. n° 5625. — GREGORIUS TURONENSIS, De virtutibus s. Martini; cf. B.H.L. n° 5618; seule l'adresse du prologue, introduite par une initiale filigranée, a été copiée au bas du f. 107v, col. b: "Domnis sanctis et in Christi amore dulcissimis fratribus... Gregorius peccator", le texte lui-même manque, soit en raison de la perte du cahier suivant, soit qu'il n'ait pas été copié. F. 108-115v. "[Passio s.] Cecilie virginis et martyris"; cf. B.H.L. Suppl. n° 1495a. F. 115v-118v. "[Passio s.] Clementis pape". [Prologus] "Postquam igitur beatus Petrus apostolus in Antiochia cathedram... -... passio secuta est"; cf. B.H.L. Suppl. n° 1849, d'après ce ms. (115v-116a, lig. 11); — "Tunc sanctus Clemens romane ecclesie episcopus disciplinam...-... Cersone Licie provincie"; C.P.L. n° 2177; B.H.L. n° 1848 (Mombritius, 2a ed., I, 341-344, var. à l'incipit et à l'explicit); suivi de: "Oremus fratres ut Dominus... participes. Per..." (116a, lig. 11-118v). F.118v-130. "[Passio] sancte Katerine virginis et martyris"; cf. B.H.L. n° 1663, sans le prologue. F. 130-130v. Additions. Table des saints contenus dans le volume, XIVe s. (130). — Prière latine en 10 strophes de deux vers, XVe s.: "Jhesu tue matris prece ab Orci me serva nece...-... ab inferi atris" (130v).

Modelagem e controle de um microveículo aéreo: uma aplicação de estabilidade robusta com a técnica backstepping em uma estrutura hexarrotor

In this Thesis, the development of the dynamic model of multirotor unmanned aerial vehicle with vertical takeoff and landing characteristics, considering input nonlinearities and a full state robust backstepping controller are presented. The dynamic model is expressed using the Newton-Euler laws, aiming to obtain a better mathematical representation of the mechanical system for system analysis and control design, not only when it is hovering, but also when it is taking-off, or landing, or flying to perform a task. The input nonlinearities are the deadzone and saturation, where the gravitational effect and the inherent physical constrains of the rotors are related and addressed. The experimental multirotor aerial vehicle is equipped with an inertial measurement unit and a sonar sensor, which appropriately provides measurements of attitude and altitude. A real-time attitude estimation scheme based on the extended Kalman filter using quaternions was developed. Then, for robustness analysis, sensors were modeled as the ideal value with addition of an unknown bias and unknown white noise. The bounded robust attitude/altitude controller were derived based on globally uniformly practically asymptotically stable for real systems, that remains globally uniformly asymptotically stable if and only if their solutions are globally uniformly bounded, dealing with convergence and stability into a ball of the state space with non-null radius, under some assumptions. The Lyapunov analysis technique was used to prove the stability of the closed-loop system, compute bounds on control gains and guaranteeing desired bounds on attitude dynamics tracking errors in the presence of measurement disturbances. The controller laws were tested in numerical simulations and in an experimental hexarotor, developed at the UFRN Robotics Laboratory

On Clifford subalgebras, spacetime splittings and applications

Z(2)-gradings of Clifford algebras are reviewed and we shall be concerned with an alpha-grading based on the structure of inner automorphisms, which is closely related to the spacetime splitting, if we consider the standard conjugation map automorphism by an arbitrary, but fixed, splitting vector. After briefly sketching the orthogonal and parallel components of products of differential forms, where we introduce the parallel [orthogonal] part as the space [time] component, we provide a detailed exposition of the Dirac operator splitting and we show how the differential operator parallel and orthogonal components are related to the Lie derivative along the splitting vector and the angular momentum splitting bivector. We also introduce multivectorial-induced alpha-gradings and present the Dirac equation in terms of the spacetime splitting, where the Dirac spinor field is shown to be a direct sum of two quaternions. We point out some possible physical applications of the formalism developed.

Geometrical logarithmic and trigonometric hypercomplex functions of quaternionic type

Quaternionic theory has greatly been developed in recent years [1-12]. Thus, in our view, the study of trigonometric and logarithmic type quaternionic functions is important for the determination and realization of a hyper complex theory. In this paper, we intend to give a geometrical foundation for both logarithmic and trigonometric hyper complex functions based on the exponential function of quaternionic type recently introduced by Borges, Marão and Machado in their paper entitled Geometrical octonions II: Hyper regularity and hyper periodicity of the exponential function appearing. © 2011 Pushpa Publishing House.

Hypercomplex geometric derivative from a Cauchy-like integral formula

The derivation and integration of hipercomplex functions have been investigated along the years, see [7], [11], [14]. The main purpose of this brief article is to give a geometrical interpretation for quaternionic derivatives, based on a recent determination of a Cauchy-like formula for quaternions, see [3]. © 2011 Academic Publications.