Quaternion Formalism Research Articles

Compensation for thermally induced birefringence in polycrystalline ceramic active elements

Polycrystalline ceramics differ significantly from single crystals in that the crystallographic axes (and hence of the axes of thermally induced birefringence) are oriented randomly in each granule of the ceramic. The quaternion formalism is employed to calculate the depolarisation in the ceramics and the efficiency of its compensation. The obtained analytic expressions are in good agreement with the numerical relations. It is shown that the larger the ratio of the sample length to the granule size, the closer the properties of the ceramics to those of a single crystal with the [111] orientation (in particular, the uncompensated depolarisation is inversely proportional to this ratio).

Transformation equations in polarization optics of inhomogeneous birefringent media.

The quaternion formalism has been used to derive new systems of equations that describe transformation of the polarization of light in inhomogeneous birefringent media. In quaternion algebra the problem of parametric representation of the unitary transformation matrix reduces to the problem of formulation of the quaternion in trigonometric form. It is shown that this can be done in 30 different ways and that to each trigonometric form corresponds its own system of transformation equations. The six simplest systems of transformation equations have been derived.

The quaternion formalism for Möbius groups in four or fewer dimensions

The Möbius group of R N ∪ {∞} defines N-dimensional inversive geometry. This geometry can serve as an alternative to projective geometry in providing a common foundation for spherical Euclidean and hyperbolic geometry. Accordingly the Möbius group plays an important role in geometry and topology. The modern emphasis on low-dimensional topology makes it timely to discuss a useful quaternion formalism for the Möbius groups in four or fewer dimensions. The present account is self-contained. It begins with the representation of quaternions by 2 x 2 matrices of complex numbers. It discusses 2 x 2 matrices of quaternions and how a suitably normalized subgroup of these matrices, extended by a certain involution related to sense reversal, is 2-1 homomorphic to the Möbius group acting on R 4 ∪ {∞}. It provides details of this action and the relation of this action to various models of the classical geometries. In higher dimensions N ⩾ 5, the best description of the Möbius group is probably by means of ( N + 2) × ( N + 2) Lorentz matrices. In the lower dimensions covered by the quaternion formalism, this alternative Lorentz formalism is a source of interesting homomorphisms. A sampling of these homomorphisms is computed explicitly both for intrinsic interest and for an illustration of the ease with which one can handle the quaternion formalism.

Representation of orientation and disorientation data for cubic, hexagonal, tetragonal and orthorhombic crystals

A convenient method for the description of orientation data for cubic, hexagonal, tetragonal and orthorhombic crystals is given. The method can also be used for the representation of disorientation data, where disorientations between any two crystals of the specified symmetry lattices are considered. It is based on the quaternion formalism introduced into the discussion of orientations and disorientations by Grimmer [Acta Cryst. (1974), A30, 685–688], Frank [(1987). Proc. Int. Conf. on Texture of Materials 8 (INCOTOM 8), Santa Fe, USA, pp. 3–13] and others. Since orientations and disorientations can be interpreted as rotations which in turn can be represented by only three parameters a vector description is used. These vectors span a rotation space corresponding to the usual space of Eulerian angles. It is called Rodrigues vector space [Rodrigues (1840). J. Math. Pure Appl. 5, 380–440; Becker & Panchanadeeswaran (1989). Text. Microstruct. 10, 167]. The direction of a Rodrigues vector is parallel to the rotation axis and its length is tan (θ/2), where θ describes the rotation angle. A method for selecting a unique representative out of the numerous symmetrically equivalent Rodrigues vectors is given. Since these selection rules depend on the symmetry of the crystal lattices considered they yield compact domains in the Rodrigues vector space which are typical for each type of lattice or lattice pair. These domains are always bounded by planes. Frank (1987) called them fundamental zones and described them for the orientations of cubic, hexagonal and orthorhombic crystals.

A Method to Calculate the g-Coefficients of the Molecular Pair Correlation Function from Molecular Dynamics Simulations

It is known that the rotational equation of motion of rigid molecules in MD simulations can be solved in a singularity-free form if quaternions are used for the description of the rotational motion. We show that these quaternions are also suited for the calculation of the so-called ‘g-coefficients’, which are the expansion coefficients of the molecular pair correlation function (MPCF) in terms of Wigner functions. This is due to the fact that quaternions arc themselves a representation of the rotation group and can be referred to an arbitrary coordinate frame in a particularly simple way. As an application for the quaternion formalism we calculate some g-coefficients of the MPCF of methylene chloride (CH2 Cl2 ).

Real Time Estimation of Aircraft Angular Attitude

This paper presents a new method of estimating aircraft angular attitude under real time conditions. The basis for the estimation is the requirement of best possible compatibility between the current estimate and the time history of the past measurements. The characteristics of the estimation technique are: (a) Use of the quaternion formalism on characterizing the rotational motion. (b) Adoption of the Least Squares (LS) method as the tool of extracting the estimated quaternion out of the foregoing requirement. (c) Incorporation of the fading memory technique as a means of reducing the sensitivity to modelling errors. The main advantage of that method is its extremely small computation burden. This is reflected by a lack of the usual operations of matrix inversion and matrix propagation in time. The paper includes: review of the pertinent quaternion properties, derivation of the estimation method and application to a real case

Quaternion-based reorientation conditions for molecular dynamics analyses

The need for a formal definition of a reorientation event in molecular dynamics simulations is recognised, and this is furnished through the quaternion formalism. Any rotation can be represented by a unit four-dimensional vector, and the general vector representing a molecule's orientation must be compared with those which represent symmetry operations. The vector dot-product is used to decide whether a given orientation is closer to the undisplaced or to the symmetry rotated orientation. When the latter occurs a reorientation event is recorded and the inverse symmetry operation is invoked. Random reorientation rates are measured by a random walk procedure, and give a basis for an objective analysis of simulation results. The reorientation conditions are then extended by the introduction of adjustable parameters in order to change the random probabilities, resulting (for instance) in the ability to identify 2-fold tetrahedral events whereas the procedure first outlined cannot recognise these events. All the relevant crystallographic rotation groups are considered, octahedral, tetrahedral, hexagonal, trigonal, tetragonal. For each of these the modified conditions are investigated.

Quaternion Approach to Rotational Compatibility of Flight Data

Abstract A new method for establishing rotational compatibility of flight data is presented. The proposed algorithm is based on a state estimation approach in which the quaternion formalism plays a major role both in characterizing the angular state and in forming the estimation technique. The method is designed for post flight analysis. Accordingly, the estimation of the rotational state at each moment is based not only on the preceding measured information but on the subsequent information as well. Particularly, the initial conditions are determined from all the time history of the rotational motion and are provided as a part of the solution. The paper includes: brief review of the quaternion formalism, description of the estimation method and, finally, application to real flight data.

Kepler orbits and the harmonic oscillator

It is shown that the Kustaanheimo-Stiefel transformation which transforms the three-dimensional Kepler problem into that for a four-dimensional harmonic oscillator may be expressed in terms of quaternions In this form the transformation represents a continuous mapping of a triad of orthonormal vectors fixed in space into a rotating triad of orthogonal vectors where one of the unit vectors is mapped into the position vector of a moving particle. The reduction of the Kepler problem to that of a harmonic oscillator is straightforward and direct in the quaternion formalism. The complex stereographic transformation used recently by the author in the corresponding reduction of Schrodinger's equation for the hydrogen atom is shown to be closely related to the quaternion representation.

The Hubble law and the spiral structures of galaxies from equations of motion in general relativity

Fully exploiting the Lie group that characterizes the underlying symmetry of general relativity theory, Einstein's tensor formalism factorizes, yielding a generalized (16-component) quaternion field formalism. The associated generalized geodesic equation, taken as the equation of motion of a star, predicts the Hubble law from one approximation for the generally covariant equations of motion, and the spiral structure of galaxies from another approximation. These results depend on the imposition of appropriate boundary conditions. The Hubble law follows when the boundary conditions derive from the oscillating model cosmology, and not from the other cosmological models. The spiral structures of the galaxies follow from the same boundary conditions, but with a different time scale than for the whole universe. The solutions that imply the spiral motion areFresnel integrals. These predict the star's motion to be along the “Cornu Spiral.” The part of this spiral in the first quadrant is the imploding phase of the galaxy, corresponding to a motion with continually decreasing radii, approaching the galactic center as time increases. The part of the “Cornu Spiral” in the third quadrant is the exploding phase, corresponding to continually increasing radii, as the star moves out from the hub. The spatial origin in the coordinate system of this curve is the inflection point, where the explosion changes to implosion. The two- (or many-) armed spiral galaxies are explained here in terms of two (or many) distinct explosions occurring at displaced times, in the domain of the rotating, planar galaxy.

Quaternion Quantum Theory: New Physics or Number Mysticism?

Hamilton's ancient quaternion formalism gave rise to vector analysis. Quaternions resurface once again to suggest relativistic generalizations of quantum theory and a new spin-12 wave equation.

Hypermass generalization of Einstein's gravitation theory

The curvilinear invariant quaternion formalism is examined for curved space time. Einstein's gravitation equation is shown to have a sample and natural form in this notation. The hypermass generalization of particle mass, which was generated in our studies of the Dirac equation, is incorporated in gravitation by generalizing Einstein's equation. Covariance requires that the gravitational constant be generalized to an invariant quaternion when the mass is. The modification appears minor and of no importance cosmologically, unless one begins considering time and mass dependence ofG.

A symmetric-Tensor-antisymmetric-tensor theory of gravitation and electromagnetism from a quaternion representation of general relativity

The factorization of Einstein’s formalism into a pair of simultaneous quaternion field equations in general relativity entails the enlargement of the original formalism from 10 to 16 independent relations. By iteration, the quaternion field equations are shown to be physically equivalent to a symmetric-tensor-antisymmetric-tensor formalism. The symmetric-tensor part is in one-to-one correspondence with the 10 relations of Einstein’s original theory of gravitation. The remaining 6 (antisymmetric tensor) relations have no counterpart in the earlier gravitational theory. In addition to the generalization of the metrical field (and therefore the description of gravitational forces) that follows from the incorporation of the antisymmetric-tensor contribution, the covariant divergence of the latter formalism automatically leads to a system of field equations whose structure is in one-to-one correspondence with the Maxwell theory for electromagnetism. It is shown that Einstein’s original formalism entails 10, rather than 16 relations, because, in addition to its covariance under the group of general relativity (aconnected topological group) it is also covariant under time and space reflection transformations. The latter is an undue restriction since it is not required by the principle of general relativity alone. When these discrete symmetry elements are dropped, the (more general) quaternion formalism results. With the expression of the latter (which is not sensitive to the «handedness» of space or the direction of time) as thesum of two formalisms—one that is even and the other odd under space or time reflections—the original 10 relations of Einstein’s equationsplus the 6 relations that lead to the Maxwell field equations follow. Finally, with the application of the Schwarzschild conditions in the anti-symmetric-tensor part of the field equations, they reduce (in orderv/c) to a scalar-field formalism. With this approximation then, the full exploitation of the principle of general relativity—in terms of a symmetric-tensor-antisymmetric-tensor formalism—reduces to a scalar-tensor formalism, such as the one that has been studied by Brans and Dicke. It follows that to within the approximations that have been considered in this comparison (which would be applicable to a derivation of planetary motion) the present theory and a trulyscalar-tensor theory would be experimentally indistinguishable. In the derivation presented in this paper, however, the generalization of the metrical field to incorporate with the usual formalism an antisymmetric-tensor part (and therefore a scalar part in the application to planetary problems) follows in a natural way from the lowest-dimensional irreducible representations of the group of general relativity and no new fundamental constants need be introduced.