Geometric Algebra Applications Vol Ii

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Geometric Algebra Applications Vol Ii

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Author : Eduardo Bayro-Corrochano
language : en
Publisher: Springer Nature
Release Date : 2020-06-19

Geometric Algebra Applications Vol Ii written by Eduardo Bayro-Corrochano and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-06-19 with Mathematics categories.

This book presents a unified mathematical treatment of diverse problems in the general domain of robotics and associated fields using Clifford or geometric alge- bra. By addressing a wide spectrum of problems in a common language, it offers both fresh insights and new solutions that are useful to scientists and engineers working in areas related with robotics. It introduces non-specialists to Clifford and geometric algebra, and provides ex- amples to help readers learn how to compute using geometric entities and geomet- ric formulations. It also includes an in-depth study of applications of Lie group theory, Lie algebra, spinors and versors and the algebra of incidence using the universal geometric algebra generated by reciprocal null cones. Featuring a detailed study of kinematics, differential kinematics and dynamics using geometric algebra, the book also develops Euler Lagrange and Hamiltoni- ans equations for dynamics using conformal geometric algebra, and the recursive Newton-Euler using screw theory in the motor algebra framework. Further, it comprehensively explores robot modeling and nonlinear controllers, and discusses several applications in computer vision, graphics, neurocomputing, quantum com- puting, robotics and control engineering using the geometric algebra framework. The book also includes over 200 exercises and tips for the development of future computer software packages for extensive calculations in geometric algebra, and a entire section focusing on how to write the subroutines in C++, Matlab and Maple to carry out efficient geometric computations in the geometric algebra framework. Lastly, it shows how program code can be optimized for real-time computations. An essential resource for applied physicists, computer scientists, AI researchers, roboticists and mechanical and electrical engineers, the book clarifies and demon- strates the importance of geometric computing for building autonomous systems to advance cognitive systems research.

Geometric Algebra Applications Vol I

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Author : Eduardo Bayro-Corrochano
language : en
Publisher: Springer
Release Date : 2018-06-20

Geometric Algebra Applications Vol I written by Eduardo Bayro-Corrochano and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-06-20 with Technology & Engineering categories.

The goal of the Volume I Geometric Algebra for Computer Vision, Graphics and Neural Computing is to present a unified mathematical treatment of diverse problems in the general domain of artificial intelligence and associated fields using Clifford, or geometric, algebra. Geometric algebra provides a rich and general mathematical framework for Geometric Cybernetics in order to develop solutions, concepts and computer algorithms without losing geometric insight of the problem in question. Current mathematical subjects can be treated in an unified manner without abandoning the mathematical system of geometric algebra for instance: multilinear algebra, projective and affine geometry, calculus on manifolds, Riemann geometry, the representation of Lie algebras and Lie groups using bivector algebras and conformal geometry. By treating a wide spectrum of problems in a common language, this Volume I offers both new insights and new solutions that should be useful to scientists, and engineers working in different areas related with the development and building of intelligent machines. Each chapter is written in accessible terms accompanied by numerous examples, figures and a complementary appendix on Clifford algebras, all to clarify the theory and the crucial aspects of the application of geometric algebra to problems in graphics engineering, image processing, pattern recognition, computer vision, machine learning, neural computing and cognitive systems.

Guide To Geometric Algebra In Practice

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Author : Leo Dorst
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-08-28

Guide To Geometric Algebra In Practice written by Leo Dorst and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-08-28 with Mathematics categories.

This highly practical Guide to Geometric Algebra in Practice reviews algebraic techniques for geometrical problems in computer science and engineering, and the relationships between them. The topics covered range from powerful new theoretical developments, to successful applications, and the development of new software and hardware tools. Topics and features: provides hands-on review exercises throughout the book, together with helpful chapter summaries; presents a concise introductory tutorial to conformal geometric algebra (CGA) in the appendices; examines the application of CGA for the description of rigid body motion, interpolation and tracking, and image processing; reviews the employment of GA in theorem proving and combinatorics; discusses the geometric algebra of lines, lower-dimensional algebras, and other alternatives to 5-dimensional CGA; proposes applications of coordinate-free methods of GA for differential geometry.

Geometric Computing With Clifford Algebras

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Author : Gerald Sommer
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29

Geometric Computing With Clifford Algebras written by Gerald Sommer and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-29 with Computers categories.

Clifford algebra, then called geometric algebra, was introduced more than a cenetury ago by William K. Clifford, building on work by Grassmann and Hamilton. Clifford or geometric algebra shows strong unifying aspects and turned out in the 1960s to be a most adequate formalism for describing different geometry-related algebraic systems as specializations of one "mother algebra" in various subfields of physics and engineering. Recent work outlines that Clifford algebra provides a universal and powerfull algebraic framework for an elegant and coherent representation of various problems occuring in computer science, signal processing, neural computing, image processing, pattern recognition, computer vision, and robotics. This monograph-like anthology introduces the concepts and framework of Clifford algebra and provides computer scientists, engineers, physicists, and mathematicians with a rich source of examples of how to work with this formalism.

Geometric Algebra Applications Vol Iii

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Author : Eduardo Bayro-Corrochano
language : en
Publisher: Springer Nature
Release Date : 2024-09-26

Geometric Algebra Applications Vol Iii written by Eduardo Bayro-Corrochano and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-09-26 with Computers categories.

The goal of Geometric Algebra Applications Vol. III: Integral Transforms, Machine Learning, and Quantum Computing is to present a unified mathematical treatment of diverse problems in the general domain like Clifford Fourier Transforms, Deep Learning and Geometric Algebra Convolutional Neural Networks, Quaternion Quantum Fourier Transform and Geometric Quantum Computing. Topics and features · Introduces nonspecialists to Clifford, or geometric algebra and by example encourages the reader to learn to compute using geometric entities and geometric formulations. · A study in depth for applications of Lie group theory, Lie algebra, projective geometry, and the algebra of incidence using the conformal geometric algebra. · Features the computing frameworks of the linear model n-dimensional affine plane and the nonlinear model of Euclidean space known as the horosphere, and addresses the relationships of these models to conformal, affine, and projective geometries. · Includes a thorough study of Integral transforms: Quaternion and Clifford Transforms, quaternion analytic signal, monogenic signals, Hilbert transform, Riesz transform, Clifford Fourier Transform, Quaternion Wavelet transforms, Quaternion Quantum Fourier Transform, 3D Radon Transform and Hough-Transform in geometric algebra. · Color image processing using the color model HSV, Quaternion Split rotors and motors, and the space-time Lorentz transform. · Geometric neural computing using Split Quaternions, Geometric Algebra neural networks, Clifford Support Vector Machine and Neuro Control. · Thorough discussion of several tasks of computer vision, graphics, neurocomputing, and robotics. machine learning, Deep Learning and CNNs, and Geometric Quantum Computing using the geometric algebra framework. · 130 exercises and hints for the development of future computer software packages for extensive calculations in geometric algebra. An entire section is dedicated to explaining how one should write the subroutines in C++, Phyton, Matlab, and Maple to carry out efficient geometric computations in the geometric algebra framework. Furthermore, it is shown how program code can be optimized for real-time computations. The book is an essential resource for applied mathematicians, physicists, computer scientists, graphics engineering, AI and Machine Learning researchers, roboticists and mechanical and electrical engineers, neurocomputing researchers, neuroscientists, and quantum computing specialists. It clarifies and demonstrates the importance of geometric computing for building autonomous systems and pushes forward advances in geometric cybernetics research.

Geometric Algebra For Computer Science

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Author : Leo Dorst
language : en
Publisher: Elsevier
Release Date : 2010-07-26

Geometric Algebra For Computer Science written by Leo Dorst and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-07-26 with Computers categories.

Until recently, almost all of the interactions between objects in virtual 3D worlds have been based on calculations performed using linear algebra. Linear algebra relies heavily on coordinates, however, which can make many geometric programming tasks very specific and complex-often a lot of effort is required to bring about even modest performance enhancements. Although linear algebra is an efficient way to specify low-level computations, it is not a suitable high-level language for geometric programming. Geometric Algebra for Computer Science presents a compelling alternative to the limitations of linear algebra. Geometric algebra, or GA, is a compact, time-effective, and performance-enhancing way to represent the geometry of 3D objects in computer programs. In this book you will find an introduction to GA that will give you a strong grasp of its relationship to linear algebra and its significance for your work. You will learn how to use GA to represent objects and perform geometric operations on them. And you will begin mastering proven techniques for making GA an integral part of your applications in a way that simplifies your code without slowing it down. * The first book on Geometric Algebra for programmers in computer graphics and entertainment computing* Written by leaders in the field providing essential information on this new technique for 3D graphics* This full colour book includes a website with GAViewer, a program to experiment with GA

Clifford Algebra To Geometric Calculus

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Author : David Hestenes
language : en
Publisher: Springer Science & Business Media
Release Date : 1984

Clifford Algebra To Geometric Calculus written by David Hestenes and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984 with Mathematics categories.

Matrix algebra has been called "the arithmetic of higher mathematics" [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebra' suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quaternions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas.

Clifford Geometric Algebras

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Author : William Baylis
language : en
Publisher: Springer Science & Business Media
Release Date : 1999-04-23

Clifford Geometric Algebras written by William Baylis and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999-04-23 with Science categories.

This volume is an outgrowth of the 1995 Summer School on Theoretical Physics of the Canadian Association of Physicists (CAP), held in Banff, Alberta, in the Canadian Rockies, from July 30 to August 12,1995. The chapters, based on lectures given at the School, are designed to be tutorial in nature, and many include exercises to assist the learning process. Most lecturers gave three or four fifty-minute lectures aimed at relative novices in the field. More emphasis is therefore placed on pedagogy and establishing comprehension than on erudition and superior scholarship. Of course, new and exciting results are presented in applications of Clifford algebras, but in a coherent and user-friendly way to the nonspecialist. The subject area of the volume is Clifford algebra and its applications. Through the geometric language of the Clifford-algebra approach, many concepts in physics are clarified, united, and extended in new and sometimes surprising directions. In particular, the approach eliminates the formal gaps that traditionally separate clas sical, quantum, and relativistic physics. It thereby makes the study of physics more efficient and the research more penetrating, and it suggests resolutions to a major physics problem of the twentieth century, namely how to unite quantum theory and gravity. The term "geometric algebra" was used by Clifford himself, and David Hestenes has suggested its use in order to emphasize its wide applicability, and b& cause the developments by Clifford were themselves based heavily on previous work by Grassmann, Hamilton, Rodrigues, Gauss, and others.

Foundations Of Geometric Algebra Computing

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Author : Dietmar Hildenbrand
language : en
Publisher: Springer
Release Date : 2015-01-29

Foundations Of Geometric Algebra Computing written by Dietmar Hildenbrand and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-01-29 with Computers categories.

The author defines “Geometric Algebra Computing” as the geometrically intuitive development of algorithms using geometric algebra with a focus on their efficient implementation, and the goal of this book is to lay the foundations for the widespread use of geometric algebra as a powerful, intuitive mathematical language for engineering applications in academia and industry. The related technology is driven by the invention of conformal geometric algebra as a 5D extension of the 4D projective geometric algebra and by the recent progress in parallel processing, and with the specific conformal geometric algebra there is a growing community in recent years applying geometric algebra to applications in computer vision, computer graphics, and robotics. This book is organized into three parts: in Part I the author focuses on the mathematical foundations; in Part II he explains the interactive handling of geometric algebra; and in Part III he deals with computing technology for high-performance implementations based on geometric algebra as a domain-specific language in standard programming languages such as C++ and OpenCL. The book is written in a tutorial style and readers should gain experience with the associated freely available software packages and applications. The book is suitable for students, engineers, and researchers in computer science, computational engineering, and mathematics.