Geometric algebra in electronics and information engineering: An introduction

Optical Mueller matrices in terms of geometric algebra

In this paper geometric algebra of space-time is used to state Maxwell's equations in an ellegant differential and integral form. The proposed integral form involves no derivative with respect to time and is coordinate-free in contrast to what is usually refered as the integral form. The Maxwell's equations are discretized in a very natural manner using their integral form. The discretization of the material equations is discussed in detail. It is directly related to the efficiency of the resulting scheme. Finally, the relation to traditional approaches is shown.

Traditional electrical power theories and one of their most important concepts—apparent power—are still a source of debate, because they present several flaws that misinterpret the power-transfer and energy-balance phenomena under distorted grid conditions. In recent years, advanced mathematical tools such as geometric algebra (GA) have been introduced to address these issues. However, the application of GA to electrical circuits requires more consensus, improvements and refinement. In this paper, electrical power theories for single-phase systems based on GA were revisited. Several drawbacks and inconsistencies of previous works were identified, and some amendments were introduced. An alternative expression is presented for the electric power in the geometric domain. Its norm is compatible with the traditional apparent power defined as the product of the RMS voltage and current. The use of this expression simplifies calculations such as those required for current decomposition. This proposal is valid even for distorted currents and voltages. Concepts are presented in a simple way so that a strong background on GA is not required. The paper included some examples and experimental results in which measurements from a utility supply were analysed.

A spatial-spectral SIFT for hyperspectral image matching and classification

In both the kinetic theory and in the theory of Brownian motion we dealt with the transition from the kinetic description in six-dimensional phase space to a more coarse description in the space of three dimensions. In Ch. 11, for example, it was the transition from the Boltzmann equation to the comparatively simple equations of gas dynamics. Then, in the theory of Brownian motion, we went over from Fokker — Planck equation for the distribution function f(r,v,t) to the simpler Einstein — Smoluchowski equation in the spatial distribution f(r,t). As indicated in Sect. 15.10, this transition corresponds to the simplified hydrodynamic description of Brownian motion

The hidden universal Clifford algebra structure of M4(R) is ambiguous in the sense that this matrix algebra is the universal geometric algebra belonging to each of the real four-dimensional quadratic vector spaces R1,3 and R2,2. As a non-universal Clifford algebra, however, M4(R) is a geometric algebra of the unique five-dimensional quadratic vector space R2,3.

The geometric algebra of a 3D Euclidean space \(G_{3,0,0}\) has a point basis and the motor algebra \(G_{3,0,1}\) a line basis. In the latter geometric algebra, the lines expressed in terms of Plucker coordinates can be used to represent points and planes as well. The reader can find a comparison of representations of points, lines, and planes using \(G_{3,0,0}\) and \(G_{3,0,1}\) in Chap. 7.

Constitutive relations in optics in terms of geometric algebra

Historically and to date, the continuity equation (C.E.) has served as a consistency criterion for the development of physical theories. In this paper, we study the C.E. employing the mathematical framework of space–time algebra (STA), showing how common equations in mathematical physics can be identified and derived from the C.E.’s structure. We show that, in STA, the nabla equation given by the geometric product between the vector derivative operator and a generalized multivector can be identified as a system of scalar and vectorial C.E.—and, thus, another form of the C.E. itself. Associated with this continuity system, decoupling conditions are determined, and a system of wave equations and the generalized analogous quantities to the energy–momentum vectors and the Lorentz force density (and their corresponding C.E.) are constructed. From the symmetry transformations that make the C.E. system’s structure invariant, a system with the structure of Maxwell’s field equations is derived. This indicates that a Maxwellian system can be derived not only from the nabla equation and the generalized continuity system as special cases, but also from the symmetries of the C.E. structure. Upon reduction to well-known simpler quantities, the results found are consistent with the usual STA treatment of electrodynamics and hydrodynamics. The diffusion equation is explored from the continuity system, where it is found that, for decoupled systems with constant or explicitly dependent diffusion coefficients, the absence of external vector sources implies a loss in the diffusion equation structure, transforming it into Helmholtz-like and wave equations.

Geometric or Clifford Algebra (CA) is a powerful mathematical tool that is attracting a growing attention in many research fields such as computer graphics, computer vision, robotics and medical imaging for its natural and intuitive way to represent geometric objects and their transformations. This paper introduces the architecture of CliffordCoreDuo, an embedded dual-core coprocessor that offers direct hardware support to four-dimensional (4D) Clifford algebra operations. A prototype implementation on an FPGA board is detailed. Experimental results show a 1.6x average speedup of CliffordCoreDuo in comparison with the baseline mono-core architecture. A potential cycle speedup of about 40x over Gaigen 2, a geometric algebra software library generator for general-purpose processors, is also demonstrated.

Geometric algebra is used in this paper for a rigorous mathematical treatment of power in single-phase circuits under nonsinusoidal conditions, as complex algebra for sinusoidal conditions. This framework clearly displays the multidimensional nature of power, which is represented by a multivector. The power multivector with its three attributes (magnitude, direction and sense) provides the means to encode all the necessary information in a single entity. This property, in conjunction with the fact that there is a one-to-one correspondence between the terms of this multivector, the instantaneous and the apparent power equation, distinguishes it as a highly efficient mathematical tool. In this way one can successfully describe power phenomena and handle practical problems (e.g., power factor improvement). Two simple examples show some of these features. In short, the power multivector under nonsinusoidal situations can be perceived as the generalization of the complex power under sinusoidal situations

Fifty years after its introduction, the perturbative approach conceived by Lax etal. [Phys. Rev. A11, 1365 (1975)10.1103/PhysRevA.11.1365] to solve Maxwell's equations remains widely used, despite its inherent divergence in the physical space. Through a vectorial analysis of the LLM algorithm carried out within the spatial frequency domain, it is here shown how each single term of the asymptotic expansion of a free-space propagated coherent light field represents the outcome of a suitable linear filter fed by the boundary field. In this way, since the Fourier transform operator naturally uncouples the roles of the boundary condition and of the propagation process, the decoding mechanism of the LLM scheme can be clearly unveiled. To this end, our main task is the exploration of the convergence features of the so-called spectral series. This is the spectral counterpart of the asymptotic series representation (in the physical space) provided by the LLM algorithm. Through a delicate asymptotic analysis, it is proved that the spectral series is uniformly convergent, within the homogeneous portion of the angular spectrum, to the Fourier transform of the coherent free-space propagator. On employing Borel summation together with analytical continuation, such a fundamental connection is then extended to the remaining evanescent part of the spectrum. In this way, it is rigorously proven that (i) exact free-space solutions of Maxwell's equations are completely encoded within their paraxial approximations and (ii) the Rayleigh-Sommerfeld coherent propagator is the generating function for all nonparaxial corrections iteratively obtained by the LLM algorithm, irrespective of the boundary condition.

This paper is about calibrating the relative poses of a group of rigidly co-located calibrated cameras. The cameras need not share any common field-of-view. A 3D test field of stationary point targets is used and the group of cameras is moved to a number of positions in the test field. The target positions need not be known exactly as they adjust during calibration. A non-linear solution is developed using geometric algebra and expressed in the conformal model of Euclidean 3D space. The algorithm is robust to outliers. It is evaluated on a large data set and results show it performs well. Some suggestions for further improvements are discussed.

The framework for playing quantum games in an Einstein-Podolsky-Rosen (EPR) type setting is investigated using the mathematical formalism of geometric algebra (GA). The main advantage of this framework is that the players' strategy sets remain identical to the ones in the classical mixed-strategy version of the game, and hence the quantum game becomes a proper extension of the classical game, avoiding a criticism of other quantum game frameworks. We produce a general solution for two-player games, and as examples, we analyze the games of Prisoners' Dilemma and Stag Hunt in the EPR setting. The use of GA allows a quantum-mechanical analysis without the use of complex numbers or the Dirac Bra-ket notation, and hence is more accessible to the non-physicist.

In this paper, I respond to a critique of one of my papers previously published in this journal, entitled `Dr. Bertlmann's socks in a quaternionic world of ambidextral reality.' The geometrical framework presented in my paper is based on a quaternionic 3-sphere, or S^3, taken as a model of the physical space in which we are inescapably confined to perform all our experiments. The framework intrinsically circumvents Bell's theorem by reproducing the singlet correlations local-realistically, without resorting to backward causation, superdeterminism, or any other conspiracy loophole. In this response, I demonstrate point by point that, contrary to its claims, the critique has not found any mistakes in my paper, either in the analytical model of the singlet correlations or in its event-by-event numerical simulation based on Geometric Algebra.