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Despite the longstanding interest in the shapes of the eggs the available parametric descriptions in the modern literature are given only via purely empirical formulas without any clear relationships with their measurable parameters. Here we present geometrical models of the eggs based on Perseus spirics and Cassinian ovals which were known since the ancient time but their analytical parameterization was also absent in the meantime. Such ones have been found recently and the present work is based on the idea to use spirics or Cassinians as geometrical models of the eggs shapes. New explicit formulas for the volumes, surface areas and the curvatures of the avian eggs have been derived from the first principles and these have been compared with the available experimental data.

Despite the longstanding interest in the shapes of the eggs since the ancient time till nowadays, the available parametric descriptions in the modern literature are given only via purely empirical formulas without any clear relationships with their measurable physical parameters. Here we present a geometrical model of the eggs based on Perseus spirics which were known as well since the ancient time but their analytical parameterizations were absent in the meantime. Such parameterizations have been found recently and the present work is based on the idea to use the spirics as a geometrical model of the egg's shapes. Explicit formulas for the volume, surface area and the curvatures of the eggs are derived from the first principles and these have been compared with the available empirical formulas and experimental data.

Dynamical orbits of the harmonic oscillator potential in the plane are ellipses which depend on a real parameter. Some time ago in this journal it has been proven by purely geometrical methods that the locus of the focuses of these ellipses are Cassinian ovals. Here we present several explicit analytic parameterizations of these remarkable curves. Nominally, their forms depend on the magnitude of the initial distance from the center of attraction and the magnitude of the initial velocity. We have found a few parameterizations in which the roles of the size and shapes can be clearly distinguished.

Despite of their importance, the symplectic groups are not so popular like orthogonal ones as they deserve. The only explanation of this fact seems to be that their algebras can not be described so simply. While in the case of the orthogonal groups they are just the anti-symmetric matrices, those of the symplectic ones should be split in four blocks that have to be specified separately. It turns out however that in some sense they can be presented by the even dimensional symmetric matrices. Here, we present such a scheme and illustrate it in the lowest possible dimension via the Cayley map. Besides, it is proved that by means of the exponential map all such matrices generate genuine symplectic matrices.

Here we demonstrate some of the benefits of a novel parameterization of the Lie groups $\mathrm{Sp}(2,\mathbb{R}) ≅ \mathrm{SL}(2,\mathbb{R})$. Relying on the properties of the exponential map $\mathfrak{sl}(2,\mathbb{R}) \rightarrow \mathrm{SL}(2,\mathbb{R})$, we have found a few explicit formulas for the logarithm of the matrices in these groups. Additionally, the explicit analytic description of the ellipse representing their field of values is derived and this allows a direct graphical identification of various types.

The present paper is a review of the research in the area of representations of the rotational motions in the three-dimensional Euclidian space. The study starts with the topics of Euler angles and Rodrigues' formula, and passes through the investigations in quaternion, spinor and vector kinematics. The authors present the interconnections between the different parameterizations of SO(3) group and stress on their merits and negative characteristics, and applications. A special attention is paid on the vector-parameterization of the rotation group, and how its nice properties are used in different mechanical applications.

We use a vector parameter description of the Lorentz groups in ℝ2,1 and ℝ3,1 to obtain an exact expression for the Thomas factor as a geometric phase. The effect of phase accumulation in Thomas-Wigner precession phenomena is seen as a manifestation of the hyperbolic solid angle theorem. On the infinitesimal level, our description involves affine connections on the noncompact Hopf fibrations U(1) → SU(1, 1) → Δ and SU(2) → PSL(2,ℂ) → H 3. The associated gauge field is a restriction of the familiar Yang-Mills anti-instanton. We also consider the dual compact case, and we discuss generalizations to arbitrary dimensions and applications in various branches of theoretical physics.

In the present paper we investigate an alternative two-axes decomposition method for rotations that has been proposed in our earlier research. It is shown to provide a convenient parametrization for many important physical systems. As an example, the kinematics of a rotating rigid body is considered and a specific class of solutions to the Euler dynamical equations are obtained in the case of symmetric inertial ellipsoid. They turn out to be related to the Rabi oscillator in spin systems well known in quantum computation. The corresponding quantum mechanical angular momentum and Laplace operator are derived as well with the aid of infinitesimal variations. Curiously, the coefficients in this new representation happen to depend only on one of the angles, which simplifies the corresponding system of ODE's emerging from separation of variables. Some applications of the hyperbolic and complex analogues of this construction in quantum mechanics and relativity are considered in a different paper cited below.

We use both vector-parameter and quaternion techniques to provide a thorough description of several classes of rotations, starting with coaxial angular velocity Ω of varying magnitude. Then, we fix the magnitude and let Ω precess at constant rate about the z-axis, which yields a particular solution to the free Euler dynamical equations in the case of axially symmetric inertial ellipsoid. The latter appears also in the description of spin precessions in NMR and quantum computing. As we show below, this problem has analytic solutions for a much larger class of motions determined by a simple condition relating the polar angle and z-projection of Ω (expressed in cylindrical coordinates), which are both time-dependent in the generic case. Relevant physical examples are also provided.

The embeddings of the $\frak{so}(3)$ Lie algebra and the Lie group ${\rm SO}(3)$ in higher dimensions is an important construction from both mathematical and physical viewpoint. Here we present results based on a program package for building the generating matrices of the irreducible embeddings of the $\frak{so}(3)$ Lie algebra within $\frak{so}(n)$ in arbitrary dimension $n \geq 3, n \neq 4k+2, k \in \mathbb{N}$ relying on the algorithm developed recently by Campoamor-Strursberg [3]. For the remaining cases $n = 4k+2$ embeddings of $\frak{so}(3)$ into $\frak{so}(n)$ are also constructed. Besides, we investigate the characteristic polynomials of these $\frak{so}(n)$ elements. We show that the Cayley map applied to $\cal{C} \in \frak{so}(n)$ is well defined and generates a subset of ${\rm SO}(n)$. Furthermore, we obtain explicit formulas for the images of the Cayley map. The so obtained ${\rm SO}(n)$ matrices are expressed as polynomials of ${\cal C}$ whose coefficients are rational functions of the norm of the vector-parameter ${\bf c}$. The composition laws are extracted for the cases $n = 4, 6$ and for the first case it is shown that via the Cayley map the isomorphism ${\rm SU}(2) \cong {\rm im}{{\rm Cay}_{{\rm im}{j_4}}}\cup\{-\mathcal{I}_4\}$ holds. Also, for $n = 4$ explicit formulas for the the angular velocity matrices are derived. Comparisons between the results obtained via the exponential map and the Cayley map are made as well. In contrast to the case of the Cayley map, the results for the exponential map include either irrational or transcendental functions of the module of the vector-parameter.

We extend an old result due to Pina on three-dimensional rotations to the hyperbolic case and utilize it to construct a specific factorization scheme for the isometries in \(\mathbb {R}^{2,1}\!\). Although somewhat restrictive compared to Euler-type decompositions, our setting allows for gimbal lock control and decouples the dependence on the compound transformation’s invariant axis and angle (rapidity), which provides convenience from both theoretical and practical point of view. In some particularly symmetric cases the solutions are obtained in a simple way from the natural parameterization of the Poincare disk. Apart from the obvious relation to \(2\!+\!1\) dimensional relativity, we discuss this approach in the context of the monodromy matrix description of quantum mechanical scattering. In the case of \(\mathbb {R}^{3,1}\) isometries, our construction is shown to generalize naturally the well-known Wigner decomposition.

Here we develop a specific factorization technique for rotations in $\mathbb{R}^3$ into five factors about two or three fixed axes. Although not always providing the most efficient solution, the method allows for avoiding gimbal lock singularities and decouples the dependence on the invariant axis ${\bf n}$ and the angle $\phi$ of the compound rotation. In particular, the solutions in the classical Euler setting are given directly by the angle of rotation $\phi$ and the coordinates of the unit vector $\bf{n}$ without additional calculations. The immediate implementations in rigid body kinematics are also discussed and some generalizations and potential applications in other branches of science and technology are pointed out as well.

On the basis of intrinsic properties of the vector parameterization of rotational motions this work presents an explicit solution of the problem of decomposition of any finite rotation into a product of three successive finite rotations about prescribed axes. The presentation of rigid body displacements is an old mechanical problem which is of great importance for solving different real tasks. In geometrical and computational mechanics the rigid body kinematics is considered in terms of vectors and matrices. In this aspect the rotation group in a three-dimensional real space may be described by combining our knowledge from analytical mechanics, vector analysis, algebra and differential geometry. Here is the place to mention that the different parameterizations of the rotation group SO(3) influence significantly the efficiency of the kinematical and dynamical models of the rigid body and multibody mechanical systems as well. Various analytical representations of rotations are obtained when the rotation is expressed by defining its action on vectors, matrices, quaternions or spinors (3). This problem is considered in detail in the review papers (2, 15) and (16). Most frequently the parameterizations of rotations are expressed via Eulerian angles as in the classical 3-1-3 case and all other combinations like 3-2-3, 2-1-2, etc., and in the case of nonrepeating axes by the Bryant-Cardan-Krylov angles. Alternatively, one can use the Euler-Rodrigues or Cayley-Klein parameters. All of them have found a lot of applications in various scientific and technological areas.

An original analytical method for casting a real three-dimensional symmetric matrix into its diagonal form along with explicit formulas for the corresponding eigenvectors is given. This is achieved by a two-step procedure relying on a nice geometrical parametrization of the rotational group SO(3) and the fundamental algebraic theorem about solutions of the polynomial equations.

In this paper we obtain the Lie derivatives of the scalar parameters in the generalized Euler decomposition with respect to arbitrary axes under left and right deck transformations. This problem can be directly related to the representation of the angular momentum in quantum mechanics. As a particular example, we calculate the angular momentum and the corresponding quantum hamiltonian in the standard Euler and Bryan representations. Similarly, in the hyperbolic case, the Laplace-Beltrami operator is retrieved for the Iwasawa decomposition. The case of two axes is considered as well.

It is well-known that a generic SO(3) transformation can be decomposed into three consecutive rotations whenever the axis of the second one is perpendicular to the other two [1]. This fact, known as the Davenport condition, provides the maximal generalization to the classical Euler setting. For arbitrary axes, on the other hand, one has a non-trivial decomposability condition [2, 3]. In the present paper we investigate the latter into more detail and use it to derive a simple set of inequalities that specify the range of the compound rotation angle, for which the condition is automatically satisfied. Thus, we obtain an estimate for the minimum number of rotations in a decomposition and provide some numerical examples.

This paper discusses the problem of composition of two arbitrary rotations in SO(3,ℝ) given in their vector-parameter form. The cases when one or two half-turns are involved in the composition or the resulting rotation is a half turn receive also rigorous treatment. Technically, this is done by lifting the problem to the covering group SU(2). Systematization of the results is made and relying on them, some applications are discussed. A convenient and economic computational method for composing rotations in vector-parameter form is proposed. It is such that the composition of two arbitrary (including half-turns) rotations can be done with at most 12 multiplications. Finally, we present an explicit variant of the construction behind Cartan’s theorem in the case of SO(3,ℝ).

This paper reviews our research activities concerning the modeling and control of rigid and elastic joint multibody mechanical systems, including some investigations into nonholonomic systems.

We propose a new method for decomposing SO(3) transformations into pairs of successive rotations (about orthogonal axes) and consider some applications in rigid body kinematics and navigation. The construction is then spread to SO(2,1) by means of Wick rotation (duality of Lie algebras) and SO(3,1) by introducing the complex vector-parameter. Possible applications in the Lorentz setting are discussed in the context of special relativity, electrodynamics and scattering theory. At the end we also consider generalizations to other six-dimensional Lie groups, namely SO(4), SO(2,2) and SO*(4).

Here we present the generalized Euler decompositions of the six-dimensional Lie groups SO(4), SO*(4) and SO(2,2) using their (local) direct product structure [1] and a technique we have developed for SO(3) and SO(2,1) (cf. [2]). Although in even dimensions the Euler invariant axis theorem is not valid, one may introduce the notion of bi-axis n ⊗ n and decompose the generalized vector-parameter c ⊗ c with respect to a given set of bi-axes. As for the Lorentz group SO(3,1), we deal with complex vector-parameters [4] and the decomposition intertwines real and imaginary parts of vectors. Thus, bi-axes in that case have the interpretation of projective lines in parameter space CP3.

In this article we suggest a new method, partially based on earlier works of Wohlhart [15], Mladenova and Mladenov [11], Brezov et al [3], that resolves the generalized Euler decomposition problem (about arbitrary axes) using a system of quadratic equations. The main contribution made here is that we manage to decouple this system and express the solutions independently in a compact covariant form. We apply the same technique to the Lorentz group in $2+1$ dimensions and discuss certain complications related to the presence of isotropic directions in $\mathbb{R}^{2,1}$.

Here we use an extension of Rodrigues' vector parameter construction for pseudo-rotations in order to obtain explicit formulae for the generalized Euler decomposition with arbitrary axes for the structure groups in the classical models of hyperbolic geometry. Although the construction is projected from the universal cover $\,\mathsf{SU}(1,1)\simeq\mathsf{SL}(2,\mathbb{R})$, most attention is paid to the 2+1 Minkowski space model, following the close analogy with the Euclidean case, and various decompositions of the restricted Lorentz group $\mathsf{SO}^+(2,1)$ are investigated in detail. At the end we propose some possible applications in special relativity and scattering theory.

We exploit a new technique for obtaining the generalized Euler decomposition of three-dimensional rotations [1], based on the vector parameter (also known as Rodrigues' or Gibbs's vector) construction in the context of the gimbal lock problem, well known in the applications. From topological point of view the latter may be thought of as a singularity of the parametrization π:RP2→T3 resulting in the loss of a degree of freedom that is believed to cause troubles in the engineering applications [2], so it is generally considered a problem and takes some effort to avoid. This article, however, focuses on its potential benefits and comments are made on how one can take advantage of them.

We use a vector parameter technique to obtain the generalized Euler decompositions with respect to arbitrarily chosen axes for the three-dimensional special orthogonal group SO(3) and the three-dimensional Lorentz group SO(2;1). Our approach, based on projecting a quaternion (respectively split quaternion) from the corresponding spin cover, has proven quite effective in various problems of geometry and physics [1, 2, 3]. In particular, we obtain explicit (generally double-valued) expressions for the three parameters in the decomposition and discuss separately the degenerate and divergent solutions, as well as decompositions with respect to two axes. There are some straightforward applications of this method in special relativity and quantum mechanics which are discussed elsewhere (see [4]).

This paper describes an approach for automatic generation of the equations of motion of elastic joint manipulators in symbolic language. It is based on a vector-parametrization of the Lie groupSO(3) and uses the Lagrange's formalism to derive the dynamical equations, the final forms of which are like the equations generated by a recursive Newton-Eulerian algorithm. These characteristics together increase the computational efficiency of the algorithm and give a very good insight into the dynamical structure of the system. In addition to this, the inertia matrix is explicitly given in the final equations, which is very important for the applicability of a mathematical model in different fields of control and simulation. The suggested algorithm is therefore quite appropriate for the purposes of robot modeling, identification as well as for the applications in real time simulations and control.

This paper treats the basic kinematical and dynamical problems of coupled rigid body systems. A vector-parametrization of the SO(3) Lie group with a manageable composition law is used and some important kinematical consequences are derived. The vector and matrix transformations are given in terms of the dual algebra and the screw geometry.

The present paper considers Hamel's equations of motion for nonholonomic systems in terms o] pseudo-coordinates and gives an efficient method (for definiteness it is called Hamel's method) for obtaining the reaction forces. The group of the nonholonomic operators is defined and its group structural constants are given. On the basis of the geometrical language it is outlined how the equations of motions are simplified when the system dynamics is considered in the tangent space of the configurational manifold. In der vorliegenden Arbeit werden die Hamelschen Bewegungsgleichungen nichtholonomer Systeme in Form von Pseudo-Koordinaten betrachtet und es wird eine effiziente Methode (die der Ubersichtlichkeit halber Hamelsche Methode genannt wird) zur Berechnung der Reaktionskrafte angegeben. Die Gruppe der nichtholonomen Operatoren wird definiert und ihre Gruppen-Struktur-Konstanten werden angegeben. Basierend auf einer geometrischen Sprache wird ausgefuhrt, wie sich die Bewegungsgleichungen vereinfachen, wenn die Systemdynamik im Tangentialraum der Konfigurationsmannigfaltigkeit betrachtet wird.

Here we derive analytic expressions for the scalar parameters which appear in the generalized Euler decomposition of the rotational matrices in $\mathbb{R}^3$. The axes of rotations in the decomposition are almost arbitrary and they need only to obey a simple condition to guarantee that the problem is well posed. A special attention is given to the case when the rotation is decomposable using only two rotations and for this case quite elegant expressions for the parameters were derived. In certain cases one encounters infinite parameters due to the rotations by an angle $\pi$ (the so called half turns). We utilize both geometric and algebraic methods to obtain those conditions that can be used to predict and deal with various configurations of that kind and then, applying l'Hôpital's rule, we easily obtain the solutions in terms of linear fractional functions. The results are summarized in two Tables and a flowchart presenting in full details the procedure.

Since a long time the group SO(n) is of a great interest in physics (space relativity theory, quantum electrodynamics, theory of elementary particles) and mechanics because of its numerous applications to problems of monitoring of unknown nonlinear systems. The present paper treats the basic theory of this group and it is shown that any transformation of the group SO(n) may be presented as a product of plane transformations in clear analytical forms, appropriate for practical applications. The approach presented here is inspired by the close analogy of plane rotations with the vector-parameterization of the SO(3) group.

In this paper we consider dynamic equations of nonholomic systes in terms of pseudo-coordinates which appear in the general theory of rigid body. Especially, the rotator kinematics and dynamics in nonholonomic coordinates is presented. Besides, using the appropriate methods of the differential geometry and theory of Lie groups and Lie algebras a simplification of motion equations in the tangent space is obtained.

Here it is shown that by coordinating any three non‐collinear points in the dynamic and the inertial Cartesian frames, measured through different technical devices, it is possible to develop via the so‐called vector‐parameter (known also as Rodrigues or Gibbs vector) both numerical and analytical algorithms for computing the position and orientation of the rigid body in the three‐dimensional Euclidean space. This study can be viewed also as an example of interesting application of the SO(3) group representations to practical problems.

In the present paper, a new formalism for obtaining the kinematical equations of a rigid body motion is proposed. It is based on an alternative parameterization of the Lie group SO(3). Here we provide a new description of attitude kinematics by means of decomposing an arbitrary rotation into a pair of successive rotations about a fixed axis u and a varying one v, which is perpendicular to u. Then u, v and u × v constitute a basis, in which it is most natural to obtain the kinematic equations. The latter appear to be quite simple when they are expressed in terms of the vector-parameter provided that the rotation about the fixed axis is applied first (in the body reference frame). The procedure is illustrated in different situations, in which explicit solutions are available. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

This paper discusses manipulator modelling and control through a nonstandard parametrization of rotation motions. The advantage of the method is the computational efficiency at the kinematic level, not from efficient presentation but from fundamental topological considerations of the configurational manifold, described by a Lie group structure.

Based on Lie groups theory, this work considers the problem of decomposition of a given rotation into three successive finite rotations with prescribed in advance axes. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

The present work aims to present a different set of parameters for Euclidean motions which have a definite physical interpretation. The orientation here is obtained via vector-parameters which are elements of a Lie group with a nice and clear composition law. The method is quite convenient for describing of rigid body and manipulators motions such as scene and image processing and recognition. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

This paper considers the dynamic equations of a rigid spacecraft under the influence of the gravity torques written as a first–order form with a constant coefficient matrix which eigenvalues are found analytically. An appropriate parameterization of SO(3) group with nice and clear geometrical properties is used and this turns out to be quite efficient in computational aspects. An example is given, which demonstrates the stability of the model. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)