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Quaternions and Cli Ord Geometric Algebras Quaternions and Cliord Geometric Algebras Robert Benjamin Easter First Draft Edition (v1) (c) copyright 2015, Robert Benjamin Easter, all rights reserved. Preface As a rst rough draft that has been put together very quickly, this book is likely to contain errata and disorganization. The references list and inline citations are very incompete, so the reader should search around for more references. I do not claim to be the inventor of any of the mathematics found here. However, some parts of this book may be considered new in some sense and were in small parts my own original research. Much of the contents was originally written by me as contributions to a web encyclopedia project just for fun, but for various reasons was inappropriate in an encyclopedic volume. I did not originally intend to write this book. This is not a dissertation, nor did its development receive any funding or proper peer review. I oer this free book to the public, such as it is, in the hope it could be helpful to an interested reader. June 19, 2015 - Robert B. Easter. (v1) [email protected] 3 Table of contents Preface . 3 List of gures . 9 1 Quaternion Algebra . 11 1.1 The Quaternion Formula . 11 1.2 The Scalar and Vector Parts . 15 1.3 The Quaternion Product . 16 1.4 The Dot Product . 16 1.5 The Cross Product . 17 1.6 Conjugates . 18 1.7 Tensor or Magnitude . 20 1.8 Versors . 20 1.9 Biradials . 22 1.10 Quaternion Identities . 23 1.11 The Biradial b/a . 25 1.12 Quadrantal Versors and Handedness . 26 1.13 Quaternions and the Left-hand Rule on Left-handed Axes . 26 1.14 Rotation Formulas . 27 1.15 The Scalar Power of a Vector . 27 1.16 Rotation of Vector Components . 28 1.17 The Product ba . 29 1.18 The Biradial a/b . 30 1.19 The Product ab . 30 1.20 Pairs of Conjugate Versors . 30 1.21 The Rotation Formula (b/a)v(a/b) . 31 1.22 The Rays a and b . 32 1.23 Reection and Conjugation . 32 1.24 Rotations as Successive Reections in Lines . 33 1.25 Equivalent Versors . 33 1.26 The Square Root of a Versor . 34 1.27 The Rodrigues Rotation Formula . 35 1.28 Rotation Around Lines and Points . 36 1.29 Vector-arcs . 38 1.30 Mean Versor . 39 1.31 Matrix Forms of Quaternions and Rotations . 46 1.32 Quaternions and Complex Numbers . 50 2 Geometric Algebra . 53 5 6 Table of contents 2.1 Quaternion Biradials in Geometric Algebra . 53 2.2 Imaginary Directed Quantities . 53 2.3 Real Directed Quantities . 54 2.4 Cliord Geometric Algebras . 54 2.5 Rules for Euclidean Vector Units . 55 2.6 The Outer Product . 56 2.7 Extended Imaginaries . 57 2.8 Scalar Multiplication . 58 2.9 Grades, Blades, and Vectors . 59 2.10 The inner product . 60 2.11 The Sum of Inner and Outer Products . 61 2.12 The reverse of a blade . 63 2.13 Reversion . 64 2.14 Conjugation of blades . 65 2.15 The square of a blade . 66 2.16 The geometric product ab of Euclidean vectors . 70 2.17 The quaternion product PQ of even-grade multivectors . 71 2.18 Dual and undual to convert to and from quaternions . 72 2.19 The inner product of quaternionic bivectors has negative sign . 72 2.20 The commutator product . 72 2.21 Evaluating expressions in Cliord geometric algebra . 74 2.22 The general form of a multivector A . 75 2.23 The algebra of the even-grade parts is quaternions . 76 2.24 Representing a quaternion versor using an odd-grade part . 77 2.25 The rotation (b/a)v(a/b) in Euclidean vectors . 78 2.26 Identities: Euclidean base unit vectors . 78 2.27 Identities: Euclidean 1-vectors . 79 2.28 Identities: The geometric product ba of 1-vectors . 79 2.29 Identities: The quaternion unit 2-vectors . 80 2.30 Quaternions mapped into a two dimensional space . 81 2.31 Identities: General form of multivector A by grade-k parts . 82 2.32 Identities: Squares, inverses, and commutative property of I . 83 2.33 Identities: Product ba and rotation operator R . 83 2.34 Identities: Vector multivector products aB and Ab . 85 2.35 Operator precedence convention for ambiguous products . 86 2.36 Identities: Projection and rejection operators for vectors . 86 2.37 Associative, distributive algebraic properties of products . 86 2.38 Recursive formulas for the inner products of blades . 87 2.39 Expansion of the geometric product of blades . 92 2.40 Denitions of products by graded parts of geometric product . 103 2.41 Identities: Inner products . 104 2.42 Duality of inner and outer products . 107 2.43 Triple vector cross product . 107 2.44 Rotation of blades . 108 2.45 Cramer's Rule and Reciprocal Bases . 109 2.45.1 Systems of scalar equations . 109 Table of contents 7 2.45.2 Systems of vector equations . 111 2.45.3 Solving for unknown vectors. 111 2.45.4 Solving for unknown scalars . 114 2.45.5 Projecting a vector onto an arbitrary basis . 114 2.45.6 Reciprocal Bases . 115 2.45.7 Tensors . 117 2.45.8 Vector Derivative Operators . 118 2.45.9 Curvilinear Coordinate Systems . 119 2.45.10 Projecting a vector onto an orthogonal basis. 120 2.45.11 Solving a system of vector equations using SymPy. 121 3 Quadric Geometric Algebra . 123 3.1 Introduction . 123 3.2 Points and GIPNS Surfaces . 124 3.2.1 QGA Point . 124 3.2.2 Inner Product Null Space . 128 3.2.3 QGA GIPNS Ellipsoid . 129 3.2.4 QGA and CGA GIPNS Sphere . 130 3.2.5 CGA 6,3D Point . 130 3.2.6 QGA and CGA GIPNS Line . 133 3.2.7 QGA and CGA GIPNS Plane . 134 3.2.8 QGA and CGA GIPNS Circle . 136 3.2.9 QGA GIPNS Cylinder . 137 3.2.10 QGA GIPNS Cone . 138 3.2.11 QGA GIPNS Elliptic Paraboloid . 139 3.2.12 QGA GIPNS Hyperbolic Paraboloid . 140 3.2.13 QGA GIPNS Hyperboloid of One Sheet . 140 3.2.14 QGA GIPNS Hyperboloid of Two Sheets . ..
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