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| geometric_algebra [2020/04/23 07:33] – [Articles] pbk | geometric_algebra [2020/04/23 07:47] – [Articles] pbk |
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| * [[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/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. | 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. |
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| | * [[https://arxiv.org/pdf/1912.05960|Geometric Algebra, Gravity and Gravitational Waves]] (2019) - //Anthony N. Lasenby// |
| | We discuss an approach to gravitational waves based on Geometric Algebra and Gauge Theory Gravity. After a brief introduction to Geometric Algebra (GA), we consider Gauge Theory Gravity, which uses symmetries expressed within the GA of flat spacetime to derive gravitational forces as the gauge forces corresponding to making these symmetries local. We then consider solutions for black holes and plane gravitational waves in this approach, noting the simplicity that GA affords in both writing the solutions, and checking some of their properties. We then go on to show that a preferred gauge emerges for gravitational plane waves, in which a `memory effect' corresponding to non-zero velocities left after the passage of the waves becomes clear, and the physical nature of this effect is demonstrated. In a final section we present the mathematical details of the gravitational wave treatment in GA, and link it with other approaches to exact waves in the literature. |
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| * [[https://vixra.org/pdf/1911.0127v1.pdf|Robust Quaternion Estimation with Geometric Algebra]] (2019) - //Mauricio C. Lopez// | * [[https://vixra.org/pdf/1911.0127v1.pdf|Robust Quaternion Estimation with Geometric Algebra]] (2019) - //Mauricio C. Lopez// |
| * [[https://arxiv.org/pdf/1911.07145|Geometric Manifolds Part I: The Directional Derivative of Scalar, Vector, Multivector, and Tensor Fields]] (2020) - //Joseph C. Schindler// | * [[https://arxiv.org/pdf/1911.07145|Geometric Manifolds Part I: The Directional Derivative of Scalar, Vector, Multivector, and Tensor Fields]] (2020) - //Joseph C. Schindler// |
| This is the first entry in a planned series aiming to establish a modified, and simpler, formalism for studying the geometry of smooth manifolds with a metric, while remaining close to standard textbook treatments in terms of notation and concepts. The key step is extending the tangent space at each point from a vector space to a geometric algebra, which is a linear space incorporating vectors with dot and wedge multiplication, and extending the affine connection to a directional derivative acting naturally on fields of multivectors (elements of the geometric algebra). (...) The theory that results from this extension is simpler and more powerful than either differential forms or tensor methods, in a number of ways. | This is the first entry in a planned series aiming to establish a modified, and simpler, formalism for studying the geometry of smooth manifolds with a metric, while remaining close to standard textbook treatments in terms of notation and concepts. The key step is extending the tangent space at each point from a vector space to a geometric algebra, which is a linear space incorporating vectors with dot and wedge multiplication, and extending the affine connection to a directional derivative acting naturally on fields of multivectors (elements of the geometric algebra). (...) The theory that results from this extension is simpler and more powerful than either differential forms or tensor methods, in a number of ways. |
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| | * [[https://arxiv.org/pdf/2002.11313|Computational Aspects of Geometric Algebra Products of Two Homogeneous Multivectors]] (2020) - //Stephane Breuils, Vincent Nozick, Akihiro Sugimoto// |
| | Studies on time and memory costs of products in geometric algebra have been limited to cases where multivectors with multiple grades have only non-zero elements. This allows to design efficient algorithms for a generic purpose; however, it does not reflect the practical usage of geometric algebra. Indeed, in applications related to geometry, multivectors are likely to be full homogeneous, having their non-zero elements over a single grade. In this paper, we provide a complete computational study on geometric algebra products of two full homogeneous multivectors, that is, the outer, inner, and geometric products of two full homogeneous multivectors. We show tight bounds on the number of the arithmetic operations required for these products. |
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| | * [[https://arxiv.org/pdf/2001.00656|Two-State Quantum Systems Revisited: A Geometric Algebra Approach]] (2020) - //Pedro Amao, Hernán Castillo// |
| | We revisit the topic of two-state quantum systems using Geometric Algebra (GA) in three dimensions G3. In this description, both the quantum states and Hermitian operators are written as elements of G3. By writing the quantum states as elements of the minimal left ideals of this algebra, we compute the energy eigenvalues and eigenvectors for the Hamiltonian of an arbitrary two-state system. The geometric interpretation of the Hermitian operators enables us to introduce an algebraic method to diagonalize these operators in GA. We then use this approach to revisit the problem of a spin-1/2 particle interacting with an external arbitrary constant magnetic field, obtaining the same results as in the conventional theory. However, GA reveals the underlying geometry of these systems, which reduces to the Larmor precession in an arbitrary plane of G3. |
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| | * [[https://arxiv.org/abs/2003.07159|Periodic Table of Geometric Numbers]] (2020) - //Garret Sobczyk// |
| | Perhaps the most significant, if not the most important, achievements in chemistry and physics are the Periodic Table of the Elements in Chemistry and the Standard Model of Elementary Particles in Physics. A comparable achievement in mathematics is the Periodic Table of Geometric Numbers discussed here. In 1878 William Kingdon Clifford discovered the defining rules for what he called geometric algebras. We show how these algebras, and their coordinate isomorphic geometric matrix algebras, fall into a natural periodic table, sidelining the superfluous definitions based upon tensor algebras and quadratic forms. |
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| * [[https://arxiv.org/pdf/2002.05993|Projective Geometric Algebra as a Subalgebra of Conformal Geometric Algebra]] (2020) - //Jaroslav Hrdina, Ales Navrat, Petr Vasik, Dietmar Hildenbrand// | * [[https://arxiv.org/pdf/2002.05993|Projective Geometric Algebra as a Subalgebra of Conformal Geometric Algebra]] (2020) - //Jaroslav Hrdina, Ales Navrat, Petr Vasik, Dietmar Hildenbrand// |
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| * [[https://arxiv.org/pdf/2004.06655|Dimensional scaffolding of electromagnetism using geometric algebra]] (2020) - //Xabier Prado Orbán, Jorge Mira// | * [[https://arxiv.org/pdf/2004.06655|Dimensional scaffolding of electromagnetism using geometric algebra]] (2020) - //Xabier Prado Orbán, Jorge Mira// |
| Using geometric algebra and calculus to express the laws of electromagnetism we are able to present magnitudes and relations in a gradual way, escalating the number of dimensions. In the one-dimensional case, charge and current densities, the electric field E and the scalar and vector potentials get a geometric interpretation in spacetime diagrams. The geometric vector derivative applied to these magnitudes yields simple expressions leading to concepts like displacement current, continuity and gauge or retarded time, with a clear geometric meaning. As the geometric vector derivative is invertible, we introduce simple Green's functions and, with this, it is possible to obtain retarded Liénard-Wiechert potentials propagating naturally at the speed of light. In two dimensions, these magnitudes become more complex, and a magnetic field B appears as a pseudoscalar which was absent in the one-dimensional world. The laws of induction reflect the relations between E and B, and it is possible to arrive to the concepts of capacitor, electric circuit and Poynting vector, explaining the flow of energy. The solutions to the wave equations in this two-dimensional scenario uncover now the propagation of physical effects at the speed of light. This anticipates the same results in the real three-dimensional world, but endowed in this case with a nature which is totally absent in one or three dimensions. Electromagnetic waves propagating entirely at the speed of light can thus be viewed as a consequence of living in a world with an odd number of spatial dimensions. Finally, in the real three-dimensional world the same set of simple multivector differential expressions encode the fundamental laws and concepts of electromagnetism. | Using geometric algebra and calculus to express the laws of electromagnetism we are able to present magnitudes and relations in a gradual way, escalating the number of dimensions. In the one-dimensional case, charge and current densities, the electric field E and the scalar and vector potentials get a geometric interpretation in spacetime diagrams. The geometric vector derivative applied to these magnitudes yields simple expressions leading to concepts like displacement current, continuity and gauge or retarded time, with a clear geometric meaning. As the geometric vector derivative is invertible, we introduce simple Green's functions and, with this, it is possible to obtain retarded Liénard-Wiechert potentials propagating naturally at the speed of light. In two dimensions, these magnitudes become more complex, and a magnetic field B appears as a pseudoscalar which was absent in the one-dimensional world. The laws of induction reflect the relations between E and B, and it is possible to arrive to the concepts of capacitor, electric circuit and Poynting vector, explaining the flow of energy. The solutions to the wave equations in this two-dimensional scenario uncover now the propagation of physical effects at the speed of light. (...) Electromagnetic waves propagating entirely at the speed of light can thus be viewed as a consequence of living in a world with an odd number of spatial dimensions. Finally, in the real three-dimensional world the same set of simple multivector differential expressions encode the fundamental laws and concepts of electromagnetism. |
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| ===== Books ===== | ===== Books ===== |
