Geometric Algebra

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This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines a product that's strongly motivated by geometry and can be taken between any two objects. For example, the product of two vectors taken in a certain way represents their common plane. This system was invented by William Clifford and is more commonly known as Clifford algebra. It's actually older than the vector algebra that we use today (due to Gibbs) and includes it as a subset. Over the years, various parts of Clifford algebra have been reinvented independently by many people who found they needed it, often not realizing that all those parts belonged in one system. This suggests that Clifford had the right idea, and that geometric algebra, not the reduced version we use today, deserves to be the standard "vector algebra." My goal in these notes is to describe geometric algebra from that standpoint and illustrate its usefulness. The notes are work in progress; I'll keep adding new topics as I learn them myself.

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... Our basic notation follows that used by [13]. We use the left-and right contractions and instead of the single dot product, our scalar product contains the reverse, A * B = AB , and our dual is a right multiplication by the pseudoscalar. ...
... Since these quite trivial examples already show many of the features we expect to encounter in more generic cases, we work them out in detail. The algebra and rules for computing the derivatives needed in this section are contained, for example, in [4,5,15,13]. ...
... This gives a method for reducing an integral on the circle to an integral on a straight line. Inserting (22) to (11) immediately yields (13). The above method is of course closely analogous to an ordinary change of variables in coordinate-based methods of integration, with the Jacobian appearing in the mapping of the pseudoscalar. ...
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We introduce a method for evaluating integrals in geometric calculus without introducing coordinates, based on using the fundamental theorem of calculus repeatedly and cutting the resulting manifolds so as to create a boundary and allow for the existence of an antiderivative at each step. The method is a direct generalization of the usual method of integration on R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document}. It may lead to both practical applications and help unveil new connections to various fields of mathematics.
... GA can be used for unified algebraic representation and manipulation of multidimensional Euclidean and non-Euclidean geometries in a consistent manner [4,5,6,7]. Many good sources exist that explain the mathematics behind GA and explore some of its possible applications [8,9,3,10,11,12,13,14,15,16,17,18,19]. These sources vary in their scope, intended audience, goals, level of details, and mathematical rigor. ...
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Designing software systems for Geometric Computing applications can be a challenging task. Software engineers typically use software abstractions to hide and manage the high complexity of such systems. Without the presence of a unifying algebraic system to describe geometric models, the use of software abstractions alone can result in many design and maintenance problems. Geometric Algebra (GA) can be a universal abstract algebraic language for software engineering geometric computing applications. Few sources, however, provide enough information about GA-based software implementations targeting the software engineering community. In particular, successfully introducing GA to software engineers requires quite different approaches from introducing GA to mathematicians or physicists. This article provides a high-level introduction to the abstract concepts and algebraic representations behind the elegant GA mathematical structure. The article focuses on the conceptual and representational abstraction levels behind GA mathematics with sufficient references for more details. In addition, the article strongly recommends applying the methods of Computational Thinking in both introducing GA to software engineers, and in using GA as a mathematical language for developing Geometric Computing software systems.
How to compare the structures of an ensemble of protein conformations is a fundamental problem in structural biology. As has been previously observed, the widely used RMSD measure due to Kabsch, in which a rigid-body superposition minimising the least-squares positional deviations is performed, has its drawbacks when comparing and visualising a set of flexible proteins structures. Here, we develop a method, fleximatch, of protein structure comparison that takes flexibility into account. Based on a distance matrix measure of flexibility, a weighted superposition of distance matrices rather than of atomic coordinates is performed. Subsequently, this allows a consistent determination of a) a superposition of structures for visualisation b) a partitioning of the protein structure into rigid molecular components (core atoms) and c) an atomic mobility measure. The method is suitable for highlighting both particularly flexible and rigid parts of a protein from structures derived from NMR, X-ray diffraction or molecular simulation. This article is protected by copyright. All rights reserved. © 2015 Wiley Periodicals, Inc.
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We give a simple, elementary, direct, and motivated construction of the geometric algebra over R n . 1.
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This chapter shows the advantages of developing the theory of linear and multilinear functions on finite dimensional spaces with Geometric Calculus. The theory is sufficiently well developed here to be readily applied to most problems of linear algebra.
Fundamentos de mecánica clásica. Contenido: Orígenes del algebra geométrica; Desarrollo del algebra geométrica; Mecánica de una partícula simple; Fuerzas centrales en un sistema de dos partículas; Operadores y trasformaciones; Sistemas de muchas partículas; Mecánica de los cuerpos rígidos; Mecánica celeste; Mecánica relativista.
  • Pertti Lounesto
  • Clifford Algebras
  • Spinors
Pertti Lounesto, Clifford Algebras and Spinors, London Mathematical Society Lecture Note Series 286, 2nd ed. (Cambridge: Cambridge University Press, 2001).