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Mathematics
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Applications of Geometric Algebra
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Applications of Geometric Algebra

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: closed (31 May 2024) | Viewed by 5103

Special Issue Editors

Prof. Dr. Francisco G. Montoya

E-Mail Website
Guest Editor
Department of Engineering, Electrical Engineering Section, University of Almeria, 04120 Almeria, Spain
Interests: geometric algebra; power quality; power theory; power engineering; optimization techniques
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Alfredo Alcayde

E-Mail Website
Guest Editor
Department of Engineering, Universidad de Almería, La Cañada de San Urbano s/n, 04120 Almería, Spain
Interests: power engineering; optimization techniques; ICT
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Geometric Algebra has emerged as a new fascinating mathematical framework for a variety of disciplines in science and engineering. At present, its application is somewhat limited, but it is increasingly being used by multiple authors in fields such as robotics, electrical engineering, image and signal processing, or deep learning to mention a few. By promoting the use of GA and its application, as well as its associated benefits, the aim is to increase its dissemination and visibility to the international scientific community for its progressive adoption as a comprehensive and universal tool.

This Special Issue invites researchers to submit original research papers and review articles related to any engineering o scientific discipline in which practical applications of Geometric Algebra are considered. The topics of interest include (but are not limited to):

  • GA in Engineering Applications: Robotics, Control, Electrical Engineering, Telecommunications, Path Planning, Optics, Material Science, Computer Graphics and Modelling,
  • GA in Applied Geometry: Spinors and Symmetry, Computation in Geometry, Molecular Geometry Furthermore, 3D Protein Structures, Computer Algebra, Curves and Surfaces,
  • GA in Information Processing: Neural Networks, Artificial Intelligence, Geographic Information Systems, Encryption and Cryptography.
  • GA in Applied Physics: Relativity, Gravity and Cosmology, Classical Physics, Electromagnetism and Optics, Quantum Physics.
  • GA in Signal, Image and Video Processing: Medical Imaging, Motion Processing, Estimation and Filtering, Features and Detection, Kernel Transformations (Fourier, Etc).
  • GA in Software: Software Libraries, Software Implementations, Software Frameworks.
  • GA in Education: Teaching GA, New Methodologies.

Prof. Dr. Francisco G. Montoya
Prof. Dr. Alfredo Alcayde
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (5 papers)

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20 pages, 2440 KiB  
Article
Conformal Image Viewpoint Invariant
by Ghina El Mir, Karim Youssef and Chady El Mir
Mathematics 2024, 12(16), 2551; https://doi.org/10.3390/math12162551 - 18 Aug 2024
Viewed by 411
Abstract
In this paper, we introduce an invariant by image viewpoint changes by applying an important theorem in conformal geometry stating that every surface of the Minkowski space R3,1 leads to an invariant by conformal transformations. For this, we identify the [...] Read more.
In this paper, we introduce an invariant by image viewpoint changes by applying an important theorem in conformal geometry stating that every surface of the Minkowski space R3,1 leads to an invariant by conformal transformations. For this, we identify the domain of an image to the disjoint union of horospheres ∐αHα of R3,1 by means of the powerful tools of the conformal Clifford algebras. We explain that every viewpoint change is given by a planar similarity and a perspective distortion encoded by the latitude angle of the camera. We model the perspective distortion by the point at infinity of the conformal model of the Euclidean plane described by D. Hestenesand we clarify the spinor representations of the similarities of the Euclidean plane. This leads us to represent the viewpoint changes by conformal transformations of ∐αHα for the Minkowski metric of the ambient space. Full article
(This article belongs to the Special Issue Applications of Geometric Algebra)
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18 pages, 353 KiB  
Article
On the Continuity Equation in Space–Time Algebra: Multivector Waves, Energy–Momentum Vectors, Diffusion, and a Derivation of Maxwell Equations
by Manuel Beato Vásquez and Melvin Arias Polanco
Mathematics 2024, 12(14), 2270; https://doi.org/10.3390/math12142270 - 20 Jul 2024
Viewed by 663
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Applications of Geometric Algebra)
19 pages, 3711 KiB  
Article
Revisiting the Hansen Problem: A Geometric Algebra Approach
by Jorge Ventura, Fernando Martinez, Isiah Zaplana, Ahmad Hosny Eid, Francisco G. Montoya and James Smith
Mathematics 2024, 12(13), 1999; https://doi.org/10.3390/math12131999 - 28 Jun 2024
Viewed by 783
Abstract
The Hansen problem is a classic and well-known geometric challenge in geodesy and surveying involving the determination of two unknown points relative to two known reference locations using angular measurements. Traditional analytical solutions rely on cumbersome trigonometric calculations and are prone to propagation [...] Read more.
The Hansen problem is a classic and well-known geometric challenge in geodesy and surveying involving the determination of two unknown points relative to two known reference locations using angular measurements. Traditional analytical solutions rely on cumbersome trigonometric calculations and are prone to propagation errors. This paper presents a novel framework leveraging geometric algebra (GA) to formulate and solve the Hansen problem. Our approach utilizes the representational capabilities of Vector Geometric Algebra (VGA) and Conformal Geometric Algebra (CGA) to avoid the need for tedious analytical manipulations and provide an efficient, unified solution. We develop concise geometric formulas tailored for computational implementation. The rigorous analyses and simulations that were completed as part of this work demonstrate that the precision and robustness of this new technique are equal or superior to those of conventional resection methods. The integration of classical concepts like the Hansen problem with modern GA-based spatial computing delivers more intuitive solutions while advancing the mathematical discourse. This work transforms conventional perspectives through methodological innovation, avoiding the limitations of prevailing paradigms. Full article
(This article belongs to the Special Issue Applications of Geometric Algebra)
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15 pages, 480 KiB  
Article
Quaternion Spin
by Bryan Sanctuary
Mathematics 2024, 12(13), 1962; https://doi.org/10.3390/math12131962 - 25 Jun 2024
Viewed by 934
Abstract
We present an analysis of the Dirac equation when the spin symmetry is changed from SU(2) to the quaternion group, Q8, achieved by multiplying one of the gamma matrices by the imaginary number, i. The reason for doing this is [...] Read more.
We present an analysis of the Dirac equation when the spin symmetry is changed from SU(2) to the quaternion group, Q8, achieved by multiplying one of the gamma matrices by the imaginary number, i. The reason for doing this is to introduce a bivector into the spin algebra, which complexifies the Dirac field. It then separates into two distinct and complementary spaces: one describing polarization and the other coherence. The former describes a 2D structured spin, and the latter its helicity, generated by a unit quaternion. Full article
(This article belongs to the Special Issue Applications of Geometric Algebra)
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18 pages, 361 KiB  
Article
Development of the Method of Averaging in Clifford Geometric Algebras
by Dmitry Shirokov
Mathematics 2023, 11(16), 3607; https://doi.org/10.3390/math11163607 - 21 Aug 2023
Viewed by 935
Abstract
We develop the method of averaging in Clifford (geometric) algebras suggested by the author in previous papers. We consider operators constructed using two different sets of anticommuting elements of real or complexified Clifford algebras. These operators generalize Reynolds operators from the representation theory [...] Read more.
We develop the method of averaging in Clifford (geometric) algebras suggested by the author in previous papers. We consider operators constructed using two different sets of anticommuting elements of real or complexified Clifford algebras. These operators generalize Reynolds operators from the representation theory of finite groups. We prove a number of new properties of these operators. Using the generalized Reynolds operators, we give a complete proof of the generalization of Pauli’s theorem to the case of Clifford algebras of arbitrary dimension. The results can be used in geometry, physics, engineering, computer science, and other applications. Full article
(This article belongs to the Special Issue Applications of Geometric Algebra)
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