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--- geometric_algebra [2020/11/02 09:23] – [Modern] pbk
+++ geometric_algebra [2020/11/19 01:14] – [Historical] pbk
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   * [[https://www.researchgate.net/publication/228955605_A_brief_introduction_to_Clifford_algebra|A brief introduction to Clifford Algebra]] (2010) - //Silvia Franchini, Giorgio Vassallo, Filippo Sorbello//
 Geometric algebra (also known as Clifford algebra) is a powerful mathematical tool that offers a natural and direct way to model geometric objects and their transformations. It is gaining growing attention in different research fields as physics, robotics, CAD/CAM and computer graphics. Clifford algebra makes geometric objects (points, lines and planes) into basic elements of computation and defines few universal operators that are applicable to all types of geometric elements. This paper provides an introduction to Clifford algebra elements and operators.
+  * [[https://vixra.org/pdf/1203.0011v1.pdf|A Very Brief Introduction to Clifford Algebra]] (2012) - //Stephen Crowley//
+This article distills many of the essential definitions from the very thorough book, Clifford Algebras: An Introduction, by Dr D.J.H. Garling, with some minor additions.
   * [[http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/bookGA.pdf|An Introduction to Geometric Algebra and Calculus]] (2014) - //Alan Bromborsky//
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 This thesis investigates the emerging field of Conformal Geometric Algebra (CGA) as a new basis for a CG framework. Computer Graphics is, fundamentally, a particular application of geometry. From a practical standpoint many of the low-level problems to do with rasterising triangles and projecting a three-dimensional world onto a computer screen have been solved and hardware especially designed for this task is available.
-  * [[http://home.deib.polimi.it/tubaro/Journals/Journal_2008_DA.pdf|3D Motion from structures of points, lines and planes]] (2007) - //Andrea Dell'Acqua, Augusto Sarti, Stefano Tubaro//
+  * [[https://tubaro.faculty.polimi.it/Journals/Journal_2008_DA.pdf|3D Motion from structures of points, lines and planes]] (2007) - //Andrea Dell'Acqua, Augusto Sarti, Stefano Tubaro//
 In this article we propose a method for estimating the camera motion from a video-sequence acquired in the presence of general 3D structures. Solutions to this problem are commonly based on the tracking of point-like features, as they usually back-project onto viewpoint-invariant 3D features. In order to improve the robustness, the accuracy and the generality of the approach, we are interested in tracking and using a wider class of structures. In addition to points, in fact, we also simultaneously consider lines and planes. In order to be able to work on all such structures with a compact and unified formalism, we use here the Conformal Model of Geometric Algebra, which proved very powerful and flexible.
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 ^                                                                                                                                           ^ Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   ^
-| [[https://archive.org/details/dieausdehnungsl04grasgoog|{{:ga:die_ausdehnungslehre_von_1844-grassmann.jpg?400}}]]                         | **Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik [Die Ausdehnungslehre von 1844] (1878)**\\ //Hermann Grassmann//\\ The Prussian schoolmaster Hermann Grassmann taught a range of subjects including mathematics, science and Latin and wrote several secondary-school textbooks. Although he was never appointed to a university post, he devoted much energy to mathematical research and developed revolutionary new insights. Die lineale Ausdehnungslehre, published in 1844, is an astonishing work which was not understood by the mathematicians of its time but which anticipated developments that took a century to come to fruition - vector spaces, dimension, exterior products and many other ideas. Admired rather than read by the next generation, it was only fully appreciated by mathematicians such as Peano and Whitehead.                                                                                                                                                |
+| [[https://archive.org/details/dieausdehnungsl04grasgoog|{{:ga:die_ausdehnungslehre_von_1844-grassmann.jpg?600}}]]                         | **Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik [Die Ausdehnungslehre von 1844] (1878)**\\ //Hermann Grassmann//\\ The Prussian schoolmaster Hermann Grassmann taught a range of subjects including mathematics, science and Latin and wrote several secondary-school textbooks. Although he was never appointed to a university post, he devoted much energy to mathematical research and developed revolutionary new insights. Die lineale Ausdehnungslehre, published in 1844, is an astonishing work which was not understood by the mathematicians of its time but which anticipated developments that took a century to come to fruition - vector spaces, dimension, exterior products and many other ideas. Admired rather than read by the next generation, it was only fully appreciated by mathematicians such as Peano and Whitehead.                                                                                                                                                |
 | [[https://archive.org/details/dieausdehnugsle00grasgoog|{{:ga:die_ausdehnungslehre-grassmann.jpg?100}}]]                                  | **Die Ausdehnungslehre (1864)**\\ //Hermann Grassmann//\\ In 1844, the Prussian schoolmaster Hermann Grassmann published Die Lineale Ausdehnungslehre. This revolutionary work anticipated the modern theory of vector spaces and exterior algebras. It was little understood at the time and the few sympathetic mathematicians, rather than trying harder to comprehend it, urged Grassmann to write an extended version of his theories. The present work is that version, first published in 1862. However, this also proved too far ahead of its time and Grassmann turned to historical linguistics, in which field his contributions are still remembered. His mathematical work eventually found champions such as Hankel, Peano, Whitehead and Élie Cartan, and it is now recognised for the brilliant achievement that it was in the history of mathematics.                                                                                                                                        |
 | [[https://archive.org/details/bub_gb_bU9rkSdWlFAC|{{:ga:theorie_der_complexen_zahlensysteme-hankel.jpg?100}}]]                            | **Theorie Der Complexen Zahlensysteme (1867)**\\ //Hermann Hankel//\\ Insbesondere der Gemeinen Imaginären Zahlen und der Hamilton'schen Quaternionen Nebst Ihrer Geometrischen Darstellung.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
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 | [[https://www.amazon.com/Advanced-Calculus-Fundamentals-Carlos-Polanco/dp/9811415072|{{:ga:advanced_calculus-polanco.jpg?100}}]] | **Advanced Calculus - Fundamentals of Mathematics (2019)**\\ // Carlos Polanco//\\ Vector calculus is an essential mathematical tool for performing mathematical analysis of physical and natural phenomena. It is employed in advanced applications in the field of engineering and computer simulations.This textbook covers the fundamental requirements of vector calculus in curricula for college students in mathematics and engineering programs. Chapters start from the basics of vector algebra, real valued functions, different forms of integrals, geometric algebra and the various theorems relevant to vector calculus and differential forms.Readers will find a concise and clear study of vector calculus, along with several examples, exercises, and a case study in each chapter.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
 | [[https://www.amazon.com/Geometric-Multiplication-Vectors-Introduction-Mathematics/dp/3030017559|{{:ga:geometric_multiplication_of_vectors-josipovic.jpg?100}}]] | **Geometric Multiplication of Vectors: An Introduction to Geometric Algebra in Physics (2019)**\\ // Miroslav Josipović//\\ Enables the reader to discover elementary concepts of geometric algebra and its applications with lucid and direct explanations. Why would one want to explore geometric algebra? What if there existed a universal mathematical language that allowed one: to make rotations in any dimension with simple formulas, to see spinors or the Pauli matrices and their products, to solve problems of the special theory of relativity in three-dimensional Euclidean space, to formulate quantum mechanics without the imaginary unit, to easily solve difficult problems of electromagnetism, to treat the Kepler problem with the formulas for a harmonic oscillator, to eliminate unintuitive matrices and tensors, to unite many branches of mathematical physics? What if it were possible to use that same framework to generalize the complex numbers or fractals to any dimension, to play with geometry on a computer, as well as to make calculations in robotics, ray-tracing and brain science? In addition, what if such a language provided a clear, geometric interpretation of mathematical objects, even for the imaginary unit in quantum mechanics? Such a mathematical language exists and it is called geometric algebra. The universality, the clear geometric interpretation, the power of generalizations to any dimension, the new insights into known theories, and the possibility of computer implementations make geometric algebra a thrilling field to unearth.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
+| [[https://www.amazon.com/dp/1704596629|{{:ga:matrix-gateway-to-geometric-algebra_sobczyk.jpg?100}}]] | **Matrix Gateway to Geometric Algebra, Spacetime and Spinors (2019)**\\ // Garret Sobczyk//\\ Geometric algebra has been presented in many different guises since its invention by William Kingdon Clifford shortly before his death in 1879. In this book we fully integrate the ideas of geometric algebra directly into the fabric of matrix linear algebra. A geometric matrix is a real or complex matrix which is identified with a unique geometric number. The matrix product of two geometric matrices is just the product of the corresponding geometric numbers. Any equation can be either interpreted as a matrix equation or an equation in geometric algebra, thus fully unifying the two languages. The first 6 chapters provide an introduction to geometric algebra, and the classification of all such algebras. The last 3 chapters explore more advanced topics in the application of geometric algebras to Pauli and Dirac spinors, special relativity, Maxwell’s equations, quaternions, split quaternions, and group manifolds. They are included to highlight the great variety of topics that are imbued with new geometric insight when expressed in geometric algebra.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
 | [[https://www.amazon.com/Clifford-Algebras-Zeons-Geometry-Combinatorics/dp/9811202575|{{:ga:clifford_algebras_and_zeons-staples.jpg?100}}]] | **Clifford Algebras And Zeons: Geometry to Combinatorics and Beyond (2020)**\\ // George Stacey Staples//\\ Clifford algebras have many well-known applications in physics, engineering, and computer graphics. Zeon algebras are subalgebras of Clifford algebras whose combinatorial properties lend them to graph-theoretic applications such as enumerating minimal cost paths in dynamic networks. This book provides a foundational working knowledge of zeon algebras, their properties, and their potential applications in an increasingly technological world. As the first textbook to explore algebraic and combinatorial properties of zeon algebras in depth, it is suitable for interdisciplinary study in analysis, algebra, and combinatorics.                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |