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--- geometric_algebra [2020/04/23 07:59] – [Articles] pbk
+++ geometric_algebra [2020/06/20 04:46] – [Articles] pbk
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   * [[https://en.wikipedia.org/wiki/Multivector|Multivector]]
   * [[https://en.wikipedia.org/wiki/Paravector|Paravector]]
+  * [[https://en.wikipedia.org/wiki/Projective_geometry|Projective Geometry]]
+  * [[https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates|Plücker coordinates]]
 ==== Personalities ====
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   * [[http://link.springer.com/content/pdf/10.1007%2Fs00006-016-0700-z.pdf|Geometric Algebra as a Unifying Language for Physics and Engineering and Its Use in the Study of Gravity]] (2016) - //Anthony Lasenby//
 Geometric Algebra (GA) is a mathematical language that aids a unified approach and understanding in topics across mathematics, physics and engineering. In this contribution, we introduce the Spacetime Algebra (STA), and discuss some of its applications in electromagnetism, quantum mechanics and acoustic physics. Then we examine a gauge theory approach to gravity that employs GA to provide a coordinate free formulation of General Relativity, and discuss what a suitable Lagrangian for gravity might look like in two dimensions. Finally the extension of the gauge theory approach to include scale invariance is briefly introduced, and attention drawn to the interesting properties with respect to the cosmological constant of the type of Lagrangians which are favoured in this approach.
+  * [[https://arxiv.org/pdf/1504.02906.pdf|Skew ray tracing in a step-index optical fiber using Geometric Algebra]] (2015) - //Angeleene S. Ang, Quirino M. Sugon, Daniel J. McNamara//
+We used Geometric Algebra to compute the paths of skew rays in a cylindrical, step-index multimode optical fiber. To do this, we used the vector addition form for the law of propagation, the exponential of an imaginary vector form for the law of refraction, and the juxtaposed vector product form for the law of reflection. In particular, the exponential forms of the vector rotations enables us to take advantage of the addition or subtraction of exponential arguments of two rotated vectors in the derivation of the ray tracing invariants in cylindrical and spherical coordinates. We showed that the light rays inside the optical fiber trace a polygonal helical path characterized by three invariants that relate successive reflections inside the fiber: the ray path distance, the difference in axial distances, and the difference in the azimuthal angles. We also rederived the known generalized formula for the numerical aperture for skew rays, which simplifies to the standard form for meridional rays.
   * [[https://arxiv.org/ftp/arxiv/papers/1502/1502.02169.pdf|Geometric algebra, qubits, geometric evolution, and all that]] (2015) - //Alexander Soiguine//
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 Recently suggested scheme of quantum computing uses g-qubit states as circular polarizations from the solution of Maxwell equations in terms of geometric algebra, along with clear definition of a complex plane as bivector in three dimensions. Here all the details of receiving the solution, and its polarization transformations are analyzed. The results can particularly be applied to the problems of quantum computing and quantum cryptography. The suggested formalism replaces conventional quantum mechanics states as objects constructed in complex vector Hilbert space framework by geometrically feasible framework of multivectors.
-  * [[https://arxiv.org/ftp/arxiv/papers/1807/1807.08603.pdf|State/observable interactions using basic geometric algebra solutions of the Maxwell equation]] (2018) - //Alexander Soiguine//
+  * [[https://arxiv.org/pdf/1807.08603|State/observable interactions using basic geometric algebra solutions of the Maxwell equation]] (2018) - //Alexander Soiguine//
 Maxwell equation in geometric algebra formalism with equally weighted basic solutions is subjected to continuously acting Clifford translation. The received states, operators acting on observables, are analyzed with different values of the Clifford translation time factor and through the observable measurement results.
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   * [[https://pastel.archives-ouvertes.fr/tel-02085820/document|Algorithmic structure for geometric algebra operators and application to quadric surfaces]] (2018) - //Stephane Breuils//
 Geometric Algebra is considered as a very intuitive tool to deal with geometric problems and it appears to be increasingly efficient and useful to deal with computer graphics problems. The Conformal Geometric Algebra includes circles, spheres, planes and lines as algebraic objects, and intersections between these objects are also algebraic objects. More complex objects such as conics, quadric surfaces can also be expressed and be manipulated using an extension of the conformal Geometric Algebra. However due to the high dimension of their representations in Geometric Algebra, implementations of Geometric Algebra that are currently available do not allow efficient realizations of these objects. In this thesis, we first present a Geometric Algebra implementation dedicated for both low and high dimensions.
+  * [[https://arxiv.org/pdf/1809.09706|Notes on Plucker's relations in Geometric Algebra]] (2018) - //Garret Sobczyk//
+Grassmannians are of fundamental importance in projective geometry, algebraic geometry, and representation theory. A vast literature has grown up utilizing using many different languages of higher mathematics, such as multilinear and tensor algebra, matroid theory, and Lie groups and Lie algebras. Here we explore the basic idea of the Plucker relations in Clifford's geometric algebra. We discover that the Plucker Relations can be fully characterized in terms of the geometric product.
   * [[https://arxiv.org/pdf/1908.08110.pdf|On the Clifford Algebraic Description of the Geometry of a 3D Euclidean Space]] (2019) - //Jaime Vaz, Stephen Mann//
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 In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex number systems. Real-part associative hypercomplex number systems generalize the well-known Cayley-Dickson algebras and real Clifford algebras and include the systems of real numbers, complex numbers, dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as particular instances. Apart from the novel hypercomplex number systems, we introduce a family of hypercomplex-valued activation functions called B-projection functions. Broadly speaking, a B-projection function projects the activation potential onto the set of all possible states of a hypercomplex-valued neuron. Using the theory presented in this paper, we confirm the stability analysis of several discrete-time hypercomplex-valued Hopfield-type neural networks from the literature. Moreover, we introduce and provide the stability analysis of a general class of Hopfield-type neural networks on Cayley-Dickson algebras.
-   * [[https://arxiv.org/pdf/1901.05873.pdf|Projective geometric algebra: A new framework for doing euclidean geometry]] (2019) - //Charles G. Gunn//
+  * [[https://arxiv.org/pdf/1903.02444|Efficient representation and manipulation of quadratic surfaces using Geometric Algebras]] (2019) - //Stéphane Breuils, Vincent Nozick, Laurent Fuchs, Akihiro Sugimoto//
+Quadratic surfaces gain more and more attention among the Geometric Algebra community and some frameworks were proposed in order to represent, transform, and intersect these quadratic surfaces. As far as the authors know, none of these frameworks support all the operations required to completely handle these surfaces. Some frameworks do not allow the construction of quadratic surfaces from control points when others do not allow to transform these quadratic surfaces. However , if we consider all the frameworks together, then all the required operations over quadratic are covered. This paper presents a unification of these frameworks that enables to represent any quadratic surfaces either using control points or from the coefficients of its implicit form. The proposed approach also allows to transform any quadratic surfaces and to compute their intersection and to easily extract some geometric properties.
+  * [[https://arxiv.org/pdf/1901.05873.pdf|Projective geometric algebra: A new framework for doing euclidean geometry]] (2019) - //Charles G. Gunn//
 A tutorial introduction to projective geometric algebra (PGA), a modern, coordinate-free framework for doing euclidean geometry. PGA features: uniform representation of points, lines, and planes; robust, parallel-safe join and meet operations; compact, polymorphic syntax for euclidean formulas and constructions; a single intuitive sandwich form for isometries; native support for automatic differentiation; and tight integration of kinematics and rigid body mechanics. Inclusion of vector, quaternion, dual quaternion, and exterior algebras as sub-algebras simplifies the learning curve and transition path for experienced practitioners. On the practical side, it can be efficiently implemented, while its rich syntax enhances programming productivity.
+  * [[https://arxiv.org/pdf/1909.02408|A Low-Memory Time-Efficient Implementation of Outermorphisms for Higher-Dimensional Geometric Algebras]] (2019) - //Ahmad Hosny Eid//
+Many important mathematical formulations in GA can be expressed as outermorphisms such as versor products, linear projection operators, and mapping between related coordinate frames. (...) This work attempts to shed some light on the problem of optimizing software implementations of outermorphisms for practical prototyping applications using geometric algebra. The approach we propose here for implementing outermorphisms requires orders of magnitude less memory compared to other common approaches, while being comparable in time performance, especially for high-dimensional geometric algebras.
   * [[http://downloads.hindawi.com/journals/cin/2019/9374802.pdf|Evaluating a Semi-Autonomous Brain-Computer Interface Based on Conformal Geometric Algebra and Artificial Vision]] (2019) - //Mauricio Ramirez-Moreno, David Gutiérrez//