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| geometric_algebra [2020/07/10 17:10] – [Articles] pbk | geometric_algebra [2020/07/19 16:33] – [Wikipedia links] pbk |
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| * [[https://en.wikipedia.org/wiki/Multivector|Multivector]] | * [[https://en.wikipedia.org/wiki/Multivector|Multivector]] |
| * [[https://en.wikipedia.org/wiki/Paravector|Paravector]] | * [[https://en.wikipedia.org/wiki/Paravector|Paravector]] |
| | * [[https://en.wikipedia.org/wiki/Outermorphism|Outermorphism]] |
| * [[https://en.wikipedia.org/wiki/Projective_geometry|Projective Geometry]] | * [[https://en.wikipedia.org/wiki/Projective_geometry|Projective Geometry]] |
| * [[https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates|Plücker coordinates]] | * [[https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates|Plücker coordinates]] |
| * [[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// |
| First we introduce briefly the frameworks of CGA and PGA for doing Euclidean geometry and we summarise basic formulas. In the next section, we show that there are actually two naturally related copies of PGA in CGA. After an identification of the two copies, the duality in PGA is obtained in terms of CGA operations. This implies directly the correspondence between flat objects and versors for Euclidean transformations in CGA and the objects and versors in PGA. | First we introduce briefly the frameworks of CGA and PGA for doing Euclidean geometry and we summarise basic formulas. In the next section, we show that there are actually two naturally related copies of PGA in CGA. After an identification of the two copies, the duality in PGA is obtained in terms of CGA operations. This implies directly the correspondence between flat objects and versors for Euclidean transformations in CGA and the objects and versors in PGA. |
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| | * [[https://link.springer.com/content/pdf/10.1007/s00006-020-1046-0.pdf|A 1d Up Approach to Conformal Geometric Algebra]] (2020) - //Anthony N. Lasenby// |
| | We discuss an alternative approach to the conformal geometric algebra (CGA) in which just a single extra dimension is necessary, as compared to the two normally used. This is made possible by working in a constant curvature background space, rather than the usual Euclidean space. A possible benefit, which is explored here, is that it is possible to define cost functions for geometric object matching in computer vision that are fully covariant, in particular invariant under both rotations and translations, unlike the cost functions which have been used in CGA so far. An algorithm is given for application of this method to the problem of matching sets of lines, which replaces the standard matrix singular value decomposition, by computations wholly in Geometric Algebra terms, and which may itself be of interest in more general settings. Secondly, we consider a further perhaps surprising application of the 1d up approach, which is to the context of a recent paper by Joy Christian published by the Royal Society, which has made strong claims about Bell's Theorem in quantum mechanics, and its relation to the sphere S^7 and the exceptional group E_8, and proposed a new associative version of the division algebra normally thought to require the octonians. |
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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. (...) 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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| | * [[https://www.ram-lab.com/papers/2020/wu2020iet.pdf|A Linear Geometric Algebra Rotor Estimator for Efficient Mesh Deformation]] (2020) - //Jin Wu, Mauricio Lopez, Ming Liu, Yilong Zhu// |
| | We solve the problem of estimating the best rotation aligning two sets of corresponding vectors (also known as Wahba's problem or point cloud registration). The proposed method is among the fastest methods reported in recent literatures, moreover it is robust to noise, accurate and simpler than most other methods. It is based on solving the linear equations derived from the formulation of the problem in Euclidean Geometric Algebra. We show its efficiency in two applications: the As-Rigid-As-Possible (ARAP) Surface Modeling and the more Smooth Rotation enhanced As-Rigid-As-Possible (SR-ARAP) mesh animation which is the only method capable of deforming surface modes with quality of tetrahedral models. Mesh deformation is a key technique in games, automated construction and robotics. |
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| * [[http://publikacio.uni-eszterhazy.hu/5004/1/AMI_online_1059.pdf|Optimized line and line segment clipping in E2 and Geometric Algebra]] (2020) - //Vaclav Skala// | * [[http://publikacio.uni-eszterhazy.hu/5004/1/AMI_online_1059.pdf|Optimized line and line segment clipping in E2 and Geometric Algebra]] (2020) - //Vaclav Skala// |
| Algorithms for line and line segment clipping are well known algorithms especially in the field of computer graphics. They are formulated for the Euclidean space representation. However, computer graphics uses the projective extension of the Euclidean space and homogeneous coordinates for representation geometric transformations with points in the E^2 or E^3 space. The projection operation from the E^3 to the E^2 space leads to the necessity to convert coordinates to the Euclidean space if the clipping operation is to be used. In this contribution, an optimized simple algorithm for line and line segment clipping in the E^2 space, which works directly with homogeneous representation and not requiring the conversion to the Euclidean space, is described. It is based on Geometric Algebra (GA) formulation for projective representation. The proposed algorithm is simple, efficient and easy to implement. | Algorithms for line and line segment clipping are well known algorithms especially in the field of computer graphics. They are formulated for the Euclidean space representation. However, computer graphics uses the projective extension of the Euclidean space and homogeneous coordinates for representation geometric transformations with points in the E^2 or E^3 space. The projection operation from the E^3 to the E^2 space leads to the necessity to convert coordinates to the Euclidean space if the clipping operation is to be used. In this contribution, an optimized simple algorithm for line and line segment clipping in the E^2 space, which works directly with homogeneous representation and not requiring the conversion to the Euclidean space, is described. It is based on Geometric Algebra (GA) formulation for projective representation. The proposed algorithm is simple, efficient and easy to implement. |
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| * [[https://www.ram-lab.com/papers/2020/wu2020iet.pdf|A linear geometric algebra rotor estimator for efficient mesh deformation]] (2020) - //Jin Wu, Mauricio Lopez, Ming Liu, Yilong Zhu// | |
| We solve the problem of estimating the best rotation aligning two sets of corresponding vectors (also known as Wahba's problem or point cloud registration). The proposed method is among the fastest methods reported in recent literatures, moreover it is robust to noise, accurate and simpler than most other methods. It is based on solving the linear equations derived from the formulation of the problem in Euclidean Geometric Algebra. We show its efficiency in two applications: the As-Rigid-As-Possible (ARAP) Surface Modeling and the more Smooth Rotation enhanced As-Rigid-As-Possible (SR-ARAP) mesh animation which is the only method capable of deforming surface modes with quality of tetrahedral models. Mesh deformation is a key technique in games, automated construction and robotics. | |
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| * [[https://arxiv.org/pdf/2007.04464.pdf|Deform, Cut and Tear a skinned model using Conformal Geometric Algebra]] (2020) - //Manos Kamarianakis, George Papagiannakis// | * [[https://arxiv.org/pdf/2007.04464.pdf|Deform, Cut and Tear a skinned model using Conformal Geometric Algebra]] (2020) - //Manos Kamarianakis, George Papagiannakis// |
