scholarly journals A Survey on the Computation of Quaternions From Rotation Matrices

2019 ◽  
Vol 11 (2) ◽  
Author(s):  
Soheil Sarabandi ◽  
Federico Thomas
Keyword(s):  
Three Dimensional ◽  
Rotation Matrix ◽  
Three Dimensions ◽  
Attitude Dynamics ◽  
Euler Parameters ◽  
Four Dimensions ◽  
Rotation Matrices ◽  
Spacecraft Attitude Dynamics ◽  
Applied Fields ◽  
Quaternion Representation

The parameterization of rotations is a central topic in many theoretical and applied fields such as rigid body mechanics, multibody dynamics, robotics, spacecraft attitude dynamics, navigation, three-dimensional image processing, and computer graphics. Nowadays, the main alternative to the use of rotation matrices, to represent rotations in ℝ3, is the use of Euler parameters arranged in quaternion form. Whereas the passage from a set of Euler parameters to the corresponding rotation matrix is unique and straightforward, the passage from a rotation matrix to its corresponding Euler parameters has been revealed to be somewhat tricky if numerical aspects are considered. Since the map from quaternions to 3 × 3 rotation matrices is a 2-to-1 covering map, this map cannot be smoothly inverted. As a consequence, it is erroneously assumed that all inversions should necessarily contain singularities that arise in the form of quotients where the divisor can be arbitrarily small. This misconception is herein clarified. This paper reviews the most representative methods available in the literature, including a comparative analysis of their computational costs and error performances. The presented analysis leads to the conclusion that Cayley's factorization, a little-known method used to compute the double quaternion representation of rotations in four dimensions from 4 × 4 rotation matrices, is the most robust method when particularized to three dimensions.

scholarly journals Nervous Activity of the Brain in Five Dimensions

Biophysica ◽  
2021 ◽  
Vol 1 (1) ◽  
pp. 38-47
Author(s):  
Arturo Tozzi ◽  
James F. Peters ◽  
Norbert Jausovec ◽  
Arjuna P. H. Don ◽  
Sheela Ramanna ◽  
...  
Keyword(s):  
Fourier Analysis ◽  
Nervous Activity ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Data Sets ◽  
Spatial And Temporal Patterns ◽  
Five Dimensions ◽  
Four Dimensions ◽  
Scale Free ◽  
The Brain

The nervous activity of the brain takes place in higher-dimensional functional spaces. It has been proposed that the brain might be equipped with phase spaces characterized by four spatial dimensions plus time, instead of the classical three plus time. This suggests that global visualization methods for exploiting four-dimensional maps of three-dimensional experimental data sets might be used in neuroscience. We asked whether it is feasible to describe the four-dimensional trajectories (plus time) of two-dimensional (plus time) electroencephalographic traces (EEG). We made use of quaternion orthographic projections to map to the surface of four-dimensional hyperspheres EEG signal patches treated with Fourier analysis. Once achieved the proper quaternion maps, we show that this multi-dimensional procedure brings undoubted benefits. The treatment of EEG traces with Fourier analysis allows the investigation the scale-free activity of the brain in terms of trajectories on hyperspheres and quaternionic networks. Repetitive spatial and temporal patterns undetectable in three dimensions (plus time) are easily enlightened in four dimensions (plus time). Further, a quaternionic approach makes it feasible to identify spatially far apart and temporally distant periodic trajectories with the same features, such as, e.g., the same oscillatory frequency or amplitude. This leads to an incisive operational assessment of global or broken symmetries, domains of attraction inside three-dimensional projections and matching descriptions between the apparently random paths hidden in the very structure of nervous fractal signals.

scholarly journals On a set of quartic surfaces in space of four dimensions; and a certain involutory transformation

1926 ◽  
Vol 110 (756) ◽  
pp. 700-708
Keyword(s):  
Present Note ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Quartic Surface ◽  
Four Dimensions ◽  
Cubic Surfaces ◽  
Mutual Relations ◽  
Quartic Surfaces ◽  
Three Dimensional Space

There exists in space of four dimensions an interesting figure of 15 lines and 15 points, first considered by Stéphanos (‘Compt. Rendus,’ vol. 93, 1881), though suggested very clearly by Cremona’s discussion of cubic surfaces in three-dimensional space. In connection with the figure of 15 lines there arises a quartic surface, the intersection of two quadrics, which is of importance as giving rise by projection to the Cyclides, as Segre has shown in detail (‘Math. Ann.,’ vol. 24, 1884). The symmetry of the figure suggests, howrever, the consideration of 15 such quartic surfaces; and it is natural to inquire as to the mutual relations of these surfaces, in particular as to their intersections. In general, two surfaces in space of four dimensions meet in a finite number of points. It appears that in this case any two of these 15 surfaces have a curve in common; it is the purpose of the present note to determine the complete intersection of any two of these 15 surfaces. Similar results may be obtained for a system of cubic surfaces in three dimensions, corresponding to those here given for this system of quartic surfaces in four dimensions, since the surfaces have one point in common, which may be taken as the centre of a projection.

The gonihedric paradigm extension of the Ising model

2015 ◽  
Vol 29 (32) ◽  
pp. 1550203 ◽  
Author(s):  
George Savvidy
Keyword(s):  
Ising Model ◽  
Spin System ◽  
Three Dimensional ◽  
Spin Systems ◽  
Three Dimensions ◽  
Partition Functions ◽  
Random Surfaces ◽  
Four Dimensions ◽  
Vacuum States ◽  
Binary Information

In this paper we review a recently suggested generalization of the Feynman path integral to an integral over random surfaces. The proposed action is proportional to the linear size of the random surfaces and is called gonihedric. The convergence and the properties of the partition function are analyzed. The model can also be formulated as a spin system with identical partition functions. The spin system represents a generalization of the Ising model with ferromagnetic, antiferromagnetic and quartic interactions. Higher symmetry of the model allows to construct dual spin systems in three and four dimensions. In three dimensions the transfer matrix describes the propagation of closed loops and we found its exact spectrum. It is a unique exact solution of the three-dimensional statistical spin system. In three and four dimensions, the system exhibits the second-order phase transitions. The gonihedric spin systems have exponentially degenerated vacuum states separated by the potential barriers and can be used as a storage of binary information.

A Multi-Axial Electromechanically-Coupled Homogenized Energy Model for Ferroelectric Materials

2017 ◽  
Author(s):  
William S. Oates ◽  
Ralph C. Smith
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Ferroelectric Materials ◽  
Energy Model ◽  
Three Dimensions ◽  
Real Time Control ◽  
Proportional Loading ◽  
Time Control ◽  
Homogenized Energy Model ◽  
Applied Fields

In this paper, we discuss the development and implementation of a 3-D electromechanically coupled homogenized energy model (HEM) for ferroelectric materials. A stochastic-based methodology is introduced and applied to problems involving large scale switching of ferroelectric and ferroelastic materials. Switching criteria for polarization variants are developed using density distributions in three dimensions to accommodate both electrical and mechanical loading and their coupled response. The theory accommodates non-proportional loading and major/minor loop hysteresis. Such formulations are known to accelerate computations for real-time control of nonlinear and hysteretic actuators. The proposed formulation maintains superior computational efficiency in the three dimensional case through the application of density formulations that are based on internal distributions of stress and electric field to produce a distribution of polarization switching events over a range of applied fields and stresses.

scholarly journals Compressible potential flows around round bodies: Janzen–Rayleigh expansion inferences

2021 ◽  
Vol 932 ◽  
Author(s):  
Idan S. Wallerstein ◽  
Uri Keshet
Keyword(s):  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Four Dimensions ◽  
Potential Flows ◽  
Prolate Spheroids ◽  
Flow Potential ◽  
Inverse Radius ◽  
Rayleigh Expansion ◽  
Simply Connected

The subsonic, compressible, potential flow around a hypersphere can be derived using the Janzen–Rayleigh expansion (JRE) of the flow potential in even powers of the incident Mach number ${\mathcal {M}}_\infty$ . JREs were carried out with terms polynomial in the inverse radius $r^{-1}$ to high orders in two dimensions, but were limited to order ${\mathcal {M}}_\infty ^{4}$ in three dimensions. We derive general JRE formulae for arbitrary order, adiabatic index and dimension. We find that powers of $\ln (r)$ can creep into the expansion, and are essential in the three-dimensional (3-D) sphere beyond order ${\mathcal {M}}_\infty ^{4}$ . Such terms are apparently absent in the 2-D disk, as we verify up to order ${\mathcal {M}}_\infty ^{100}$ , although they do appear in other dimensions (e.g. at order ${\mathcal {M}}_\infty ^{2}$ in four dimensions). An exploration of various 2-D and 3-D bodies suggests a topological connection, with logarithmic terms emerging when the flow is simply connected. Our results have additional physical implications. They are used to improve the hodograph-based approximation for the flow in front of a sphere. The symmetry-axis velocity profiles of axisymmetric flows around different prolate spheroids are approximately related to each other by a simple, Mach-independent scaling.

scholarly journals Incidences with Curves in $\mathbb{R}^d$

10.37236/5840 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Micha Sharir ◽  
Adam Sheffer ◽  
Noam Solomon
Keyword(s):  
Recent Result ◽  
Degrees Of Freedom ◽  
Algebraic Curves ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Bounded Degree ◽  
Four Dimensions ◽  
Mild Conditions ◽  
Point Line ◽  
Special Case

We prove that the number of incidences between $m$ points and $n$ bounded-degree curves with $k$ degrees of freedom in ${\mathbb R}^d$ is\[ O\left(m^{\frac{k}{dk-d+1}+\varepsilon}n^{\frac{dk-d}{dk-d+1}}+ \sum_{j=2}^{d-1} m^{\frac{k}{jk-j+1}+\varepsilon}n^{\frac{d(j-1)(k-1)}{(d-1)(jk-j+1)}} q_j^{\frac{(d-j)(k-1)}{(d-1)(jk-j+1)}}+m+n\right),\]for any $\varepsilon>0$, where the constant of proportionality depends on $k, \varepsilon$ and $d$, provided that no $j$-dimensional surface of degree $\le c_j(k,d,\varepsilon)$, a constant parameter depending on $k$, $d$, $j$, and $\varepsilon$, contains more than $q_j$ input curves, and that the $q_j$'s satisfy certain mild conditions. This bound generalizes the well-known planar incidence bound of Pach and Sharir to $\mathbb{R}^d$. It generalizes a recent result of Sharir and Solomon concerning point-line incidences in four dimensions (where d=4 and k=2), and partly generalizes a recent result of Guth (as well as the earlier bound of Guth and Katz) in three dimensions (Guth's three-dimensional bound has a better dependency on $q_2$). It also improves a recent d-dimensional general incidence bound by Fox, Pach, Sheffer, Suk, and Zahl, in the special case of incidences with algebraic curves. Our results are also related to recent works by Dvir and Gopi and by Hablicsek and Scherr concerning rich lines in high-dimensional spaces. Our bound is not known to be tight in most cases.

scholarly journals The relations connecting the angle-sums and volume of a polytope in space of n dimensions

1927 ◽  
Vol 115 (770) ◽  
pp. 103-119 ◽  
Keyword(s):  
Three Dimensional ◽  
Spherical Geometry ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Euler Formula ◽  
Five Dimensions ◽  
Four Dimensions ◽  
Elliptic Geometry ◽  
Space Constant ◽  
Different Dimensions

1.1. In two-dimensional spherical or elliptic geometry we have the familiar relation between the area of a triangle and its angle-sum, ∆ = k 2 (∑ α ‒ π ), (1.11) 2 π being the measure of the whole angle at a point, and k the space-constant, or, in spherical geometry, the radius of the sphere. It is well known that in three-dimensional spherical or elliptic geometry there is no corresponding relation involving the volume of a tetrahedron. For elliptic or hyperbolic space of four dimensions it was proved by Dehn that the volume of a simplex can be expressed linearly in terms of the sums of the dihedral angles (angles at a face), angles at an edge, and angles at a vertex, but for space of five dimensions the linear relations do not involve the volume. He indicates also, in a general way, the extensions of these results for spaces of any odd or even dimensions. He shows further that these results are connected with the form of the Euler polyhedral theorem, which is expressed by a linear relation connecting the numbers of boundaries of different dimensions, and which for space of odd dimensions is not homogeneous, e. g ., N 2 — N 1 + N 0 = 2 in three dimen­sions, but for space of even dimensions is homogeneous, e. g ., N 1 — N 0 = 0 in two dimensions, N 3 — N 2 + N 1 — N 0 = 0 in four. The connection, as Dehn points out, was made use of by Legendre in a proof which he gave for the Euler formula in three dimensions. 1.2. Dehn extends this connection in detail for four and five dimensions, and states the following general results in space of dimensions R n for simplexes and for polytopes bounded entirely by simplexes : (1) In R n there are ½ n + 1 or ½( n + 1) relations (according as n is even or odd) between the numbers of boundaries of a polytope bounded by simplexes.

NON-LINEAR SURFACE SUPERCONDUCTIVITY IN THREE DIMENSIONS IN THE LARGE κ LIMIT

2004 ◽  
Vol 06 (04) ◽  
pp. 637-652 ◽  
Author(s):  
Y. ALMOG
Keyword(s):  
Order Parameter ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Bounded Domains ◽  
Ginzburg Landau ◽  
Landau Model ◽  
Surface Superconductivity ◽  
Bulk Superconductivity ◽  
Decay Condition ◽  
Applied Fields

The Ginzburg–Landau model of superconductivity is considered in three dimensions. We show, for smooth bounded domains, that the superconductivity order parameter decays exponentially fast away from the boundary as the Ginzburg–Landau parameter κ tends to infinity. We prove this result for applied magnetic fields satisfying |hex|-κ≫κ1/2. Additionally, we prove that for applied fields greater than HC2, the only solution in ℝ3 satisfying a certain decay condition is the normal state. Finally we prove that bulk superconductivity decreases to zero as |hex|↑HC2, and thus extend (though in a weaker sense), the results in [23] to three-dimensional settings.

scholarly journals Effective theories and non-minimal couplings in low-dimensional systems

2021 ◽  
pp. 2150099
Author(s):  
M. G. Campos ◽  
L. P. R. Ospedal
Keyword(s):  
Three Dimensional ◽  
Three Dimensions ◽  
Minimal Coupling ◽  
Four Dimensions ◽  
Dimensional Spacetime ◽  
External Electromagnetic Fields ◽  
Fermionic Fields ◽  
Low Dimensional ◽  
Effective Theories ◽  
Chern Simons

Dimensionality aspects of non-minimal electromagnetic couplings are investigated. By means of the Foldy–Wouthuysen transformation, we attain (non-)relativistic interactions related to the non-minimal coupling in three-dimensional spacetime, for both the bosonic and fermionic fields. Next, we establish some comparisons and analyze particular situations in which the external electromagnetic fields are described either by Maxwell or Maxwell–Chern–Simons Electrodynamics. In addition, we consider the situation of a non-minimal coupling for the fermionic field in four dimensions, carry out its dimensional reduction to three dimensions and show that the three-dimensional scenario previously worked out can be recovered as a particular case. Finally, we discuss a number of structural aspects of both procedures.

THE FREEDMAN-TOWNSEND MODEL IN TWO AND THREE DIMENSIONS

1992 ◽  
Vol 07 (09) ◽  
pp. 2005-2020 ◽  
Author(s):  
D.G.C. MCKEON
Keyword(s):  
Configuration Space ◽  
Three Dimensional ◽  
Transverse Component ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Nonlinear Sigma Model ◽  
Four Dimensions ◽  
Space Formalism ◽  
Vector Theory ◽  
Definition Of

Using a mechanism similar to that employed by Freedman and Townsend in four dimensions, we discuss a variety of two- and three-dimensional gauge theories. The simplest of these models is equivalent to the nonlinear sigma model; another corresponds to a massive vector theory. In three dimensions, there exists a gauge invariance associated with the auxiliary vector ϕμa; when we quantize, this is accommodated using both the Batalin-Vilkovisky configuration space formalism and the Batalin-Fradkin-Vilkovisky configuration space formalism. Explicit one-loop calculations in the simplest two-dimensional model are carried out. The regularization used is a variant of operator regularization, allowing one to remain in two dimensions, hence circumventing problems associated with definition of the antisymmetric tensor εμν. This model is renormalizable, with renormalization mixing the scalar field ϕa and the transverse component of the vector field Vμa.

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