Università Degli Studi Di Trieste a Gentle Introduction to Clifford Algebra
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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 . -
Geometric Algebra Rotors for Skinned Character Animation Blending
Geometric algebra rotors for skinned character animation blending briefs_0080* QLB: 320 FPS GA: 381 FPS QLB: 301 FPS GA: 341 FPS DQB: 290 FPS GA: 325 FPS Figure 1: Comparison between animation blending techniques for skinned characters with variable complexity: a) quaternion linear blending (QLB) and dual-quaternion slerp-based interpolation (DQB) during real-time rigged animation, and b) our faster geometric algebra (GA) rotors in Euclidean 3D space as a first step for further character-simulation related operations and transformations. We employ geometric algebra as a single algebraic framework unifying previous separate linear and (dual) quaternion algebras. Abstract 2 Previous work The main goal and contribution of this work is to show that [McCarthy 1990] has already illustrated the connection between (automatically generated) computer implementations of geometric quaternions and GA bivectors as well as the different Clifford algebra (GA) can perform at a faster level compared to standard algebras with degenerate scalar products that can be used to (dual) quaternion geometry implementations for real-time describe dual quaternions. Even simple quaternions are identified character animation blending. By this we mean that if some piece as 3-D Euclidean taken out of their propert geometric algebra of geometry (e.g. Quaternions) is implemented through geometric context. algebra, the result is as efficient in terms of visual quality and even faster (in terms of computation time and memory usage) as [Fontijne and Dorst 2003] and [Dorst et al. 2007] have illustrated the traditional quaternion and dual quaternion algebra the use of all three GA models with applications from computer implementation. This should be so even without taking into vision, animation as well as a basic recursive ray-tracer. -
21. Orthonormal Bases
21. Orthonormal Bases The canonical/standard basis 011 001 001 B C B C B C B0C B1C B0C e1 = B.C ; e2 = B.C ; : : : ; en = B.C B.C B.C B.C @.A @.A @.A 0 0 1 has many useful properties. • Each of the standard basis vectors has unit length: q p T jjeijj = ei ei = ei ei = 1: • The standard basis vectors are orthogonal (in other words, at right angles or perpendicular). T ei ej = ei ej = 0 when i 6= j This is summarized by ( 1 i = j eT e = δ = ; i j ij 0 i 6= j where δij is the Kronecker delta. Notice that the Kronecker delta gives the entries of the identity matrix. Given column vectors v and w, we have seen that the dot product v w is the same as the matrix multiplication vT w. This is the inner product on n T R . We can also form the outer product vw , which gives a square matrix. 1 The outer product on the standard basis vectors is interesting. Set T Π1 = e1e1 011 B C B0C = B.C 1 0 ::: 0 B.C @.A 0 01 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 0 . T Πn = enen 001 B C B0C = B.C 0 0 ::: 1 B.C @.A 1 00 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 1 In short, Πi is the diagonal square matrix with a 1 in the ith diagonal position and zeros everywhere else. -
Exercise 1 Advanced Sdps Matrices and Vectors
Exercise 1 Advanced SDPs Matrices and vectors: All these are very important facts that we will use repeatedly, and should be internalized. • Inner product and outer products. Let u = (u1; : : : ; un) and v = (v1; : : : ; vn) be vectors in n T Pn R . Then u v = hu; vi = i=1 uivi is the inner product of u and v. T The outer product uv is an n × n rank 1 matrix B with entries Bij = uivj. The matrix B is a very useful operator. Suppose v is a unit vector. Then, B sends v to u i.e. Bv = uvT v = u, but Bw = 0 for all w 2 v?. m×n n×p • Matrix Product. For any two matrices A 2 R and B 2 R , the standard matrix Pn product C = AB is the m × p matrix with entries cij = k=1 aikbkj. Here are two very useful ways to view this. T Inner products: Let ri be the i-th row of A, or equivalently the i-th column of A , the T transpose of A. Let bj denote the j-th column of B. Then cij = ri cj is the dot product of i-column of AT and j-th column of B. Sums of outer products: C can also be expressed as outer products of columns of A and rows of B. Pn T Exercise: Show that C = k=1 akbk where ak is the k-th column of A and bk is the k-th row of B (or equivalently the k-column of BT ). -
The Construction of Spinors in Geometric Algebra
The Construction of Spinors in Geometric Algebra Matthew R. Francis∗ and Arthur Kosowsky† Dept. of Physics and Astronomy, Rutgers University 136 Frelinghuysen Road, Piscataway, NJ 08854 (Dated: February 4, 2008) The relationship between spinors and Clifford (or geometric) algebra has long been studied, but little consistency may be found between the various approaches. However, when spinors are defined to be elements of the even subalgebra of some real geometric algebra, the gap between algebraic, geometric, and physical methods is closed. Spinors are developed in any number of dimensions from a discussion of spin groups, followed by the specific cases of U(1), SU(2), and SL(2, C) spinors. The physical observables in Schr¨odinger-Pauli theory and Dirac theory are found, and the relationship between Dirac, Lorentz, Weyl, and Majorana spinors is made explicit. The use of a real geometric algebra, as opposed to one defined over the complex numbers, provides a simpler construction and advantages of conceptual and theoretical clarity not available in other approaches. I. INTRODUCTION Spinors are used in a wide range of fields, from the quantum physics of fermions and general relativity, to fairly abstract areas of algebra and geometry. Independent of the particular application, the defining characteristic of spinors is their behavior under rotations: for a given angle θ that a vector or tensorial object rotates, a spinor rotates by θ/2, and hence takes two full rotations to return to its original configuration. The spin groups, which are universal coverings of the rotation groups, govern this behavior, and are frequently defined in the language of geometric (Clifford) algebras [1, 2]. -
The Devil of Rotations Is Afoot! (James Watt in 1781)
The Devil of Rotations is Afoot! (James Watt in 1781) Leo Dorst Informatics Institute, University of Amsterdam XVII summer school, Santander, 2016 0 1 The ratio of vectors is an operator in 2D Given a and b, find a vector x that is to c what b is to a? So, solve x from: x : c = b : a: The answer is, by geometric product: x = (b=a) c kbk = cos(φ) − I sin(φ) c kak = ρ e−Iφ c; an operator on c! Here I is the unit-2-blade of the plane `from a to b' (so I2 = −1), ρ is the ratio of their norms, and φ is the angle between them. (Actually, it is better to think of Iφ as the angle.) Result not fully dependent on a and b, so better parametrize by ρ and Iφ. GAViewer: a = e1, label(a), b = e1+e2, label(b), c = -e1+2 e2, dynamicfx = (b/a) c,g 1 2 Another idea: rotation as multiple reflection Reflection in an origin plane with unit normal a x 7! x − 2(x · a) a=kak2 (classic LA): Now consider the dot product as the symmetric part of a more fundamental geometric product: 1 x · a = 2(x a + a x): Then rewrite (with linearity, associativity): x 7! x − (x a + a x) a=kak2 (GA product) = −a x a−1 with the geometric inverse of a vector: −1 2 FIG(7,1) a = a=kak . 2 3 Orthogonal Transformations as Products of Unit Vectors A reflection in two successive origin planes a and b: x 7! −b (−a x a−1) b−1 = (b a) x (b a)−1 So a rotation is represented by the geometric product of two vectors b a, also an element of the algebra. -
Geometric (Clifford) Algebra and Its Applications
Geometric (Clifford) algebra and its applications Douglas Lundholm F01, KTH January 23, 2006∗ Abstract In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and opera- tions. This view enables us to define Clifford algebras with scalars in arbitrary rings and provides new suggestions for an infinite-dimensional approach. Furthermore, I give a quick review of classic results regarding geo- metric algebras, such as their classification in terms of matrix algebras, the connection to orthogonal and Spin groups, and their representation theory. A number of lower-dimensional examples are worked out in a sys- tematic way using so called norm functions, while general applications of representation theory include normed division algebras and vector fields on spheres. I also consider examples in relativistic physics, where reformulations in terms of geometric algebra give rise to both computational and conceptual simplifications. arXiv:math/0605280v1 [math.RA] 10 May 2006 ∗Corrected May 2, 2006. Contents 1 Introduction 1 2 Foundations 3 2.1 Geometric algebra ( , q)...................... 3 2.2 Combinatorial CliffordG V algebra l(X,R,r)............. 6 2.3 Standardoperations .........................C 9 2.4 Vectorspacegeometry . 13 2.5 Linearfunctions ........................... 16 2.6 Infinite-dimensional Clifford algebra . 19 3 Isomorphisms 23 4 Groups 28 4.1 Group actions on .......................... 28 4.2 TheLipschitzgroupG ......................... 30 4.3 PropertiesofPinandSpingroups . 31 4.4 Spinors ................................ 34 5 A study of lower-dimensional algebras 36 5.1 (R1) ................................. 36 G 5.2 (R0,1) =∼ C -Thecomplexnumbers . 36 5.3 G(R0,0,1)............................... -
A Guided Tour to the Plane-Based Geometric Algebra PGA
A Guided Tour to the Plane-Based Geometric Algebra PGA Leo Dorst University of Amsterdam Version 1.15{ July 6, 2020 Planes are the primitive elements for the constructions of objects and oper- ators in Euclidean geometry. Triangulated meshes are built from them, and reflections in multiple planes are a mathematically pure way to construct Euclidean motions. A geometric algebra based on planes is therefore a natural choice to unify objects and operators for Euclidean geometry. The usual claims of `com- pleteness' of the GA approach leads us to hope that it might contain, in a single framework, all representations ever designed for Euclidean geometry - including normal vectors, directions as points at infinity, Pl¨ucker coordinates for lines, quaternions as 3D rotations around the origin, and dual quaternions for rigid body motions; and even spinors. This text provides a guided tour to this algebra of planes PGA. It indeed shows how all such computationally efficient methods are incorporated and related. We will see how the PGA elements naturally group into blocks of four coordinates in an implementation, and how this more complete under- standing of the embedding suggests some handy choices to avoid extraneous computations. In the unified PGA framework, one never switches between efficient representations for subtasks, and this obviously saves any time spent on data conversions. Relative to other treatments of PGA, this text is rather light on the mathematics. Where you see careful derivations, they involve the aspects of orientation and magnitude. These features have been neglected by authors focussing on the mathematical beauty of the projective nature of the algebra. -
Quantum/Classical Interface: Fermion Spin
Quantum/Classical Interface: Fermion Spin W. E. Baylis, R. Cabrera, and D. Keselica Department of Physics, University of Windsor Although intrinsic spin is usually viewed as a purely quantum property with no classical ana- log, we present evidence here that fermion spin has a classical origin rooted in the geometry of three-dimensional physical space. Our approach to the quantum/classical interface is based on a formulation of relativistic classical mechanics that uses spinors. Spinors and projectors arise natu- rally in the Clifford’s geometric algebra of physical space and not only provide powerful tools for solving problems in classical electrodynamics, but also reproduce a number of quantum results. In particular, many properites of elementary fermions, as spin-1/2 particles, are obtained classically and relate spin, the associated g-factor, its coupling to an external magnetic field, and its magnetic moment to Zitterbewegung and de Broglie waves. The relationship is further strengthened by the fact that physical space and its geometric algebra can be derived from fermion annihilation and creation operators. The approach resolves Pauli’s argument against treating time as an operator by recognizing phase factors as projected rotation operators. I. INTRODUCTION The intrinsic spin of elementary fermions like the electron is traditionally viewed as an essentially quantum property with no classical counterpart.[1, 2] Its two-valued nature, the fact that any measurement of the spin in an arbitrary direction gives a statistical distribution of either “spin up” or “spin down” and nothing in between, together with the commutation relation between orthogonal components, is responsible for many of its “nonclassical” properties. -
Inverse and Determinant in 0 to 5 Dimensional Clifford Algebra
Clifford Algebra Inverse and Determinant Inverse and Determinant in 0 to 5 Dimensional Clifford Algebra Peruzan Dadbeha) (Dated: March 15, 2011) This paper presents equations for the inverse of a Clifford number in Clifford algebras of up to five dimensions. In presenting these, there are also presented formulas for the determinant and adjugate of a general Clifford number of up to five dimensions, matching the determinant and adjugate of the matrix representations of the algebra. These equations are independent of the metric used. PACS numbers: 02.40.Gh, 02.10.-v, 02.10.De Keywords: clifford algebra, geometric algebra, inverse, determinant, adjugate, cofactor, adjoint I. INTRODUCTION the grade. The possible grades are from the grade-zero scalar and grade-1 vector elements, up to the grade-(d−1) Clifford algebra is one of the more useful math tools pseudovector and grade-d pseudoscalar, where d is the di- for modeling and analysis of geometric relations and ori- mension of Gd. entations. The algebra is structured as the sum of a Structurally, using the 1–dimensional ei’s, or 1-basis, scalar term plus terms of anticommuting products of vec- to represent Gd involves manipulating the products in tors. The algebra itself is independent of the basis used each term of a Clifford number by using the anticom- for computations, and only depends on the grades of muting part of the Clifford product to move the 1-basis the r-vector parts and their relative orientations. It is ei parts around in the product, as well as using the inner- this implementation (demonstrated via the standard or- product to eliminate the ei pairs with their metric equiv- thonormal representation) that will be used in the proofs. -
Geometric Algebra 4
Geometric Algebra 4. Algebraic Foundations and 4D Dr Chris Doran ARM Research L4 S2 Axioms Elements of a geometric algebra are Multivectors can be classified by grade called multivectors Grade-0 terms are real scalars Grading is a projection operation Space is linear over the scalars. All simple and natural L4 S3 Axioms The grade-1 elements of a geometric The antisymmetric produce of r vectors algebra are called vectors results in a grade-r blade Call this the outer product So we define Sum over all permutations with epsilon +1 for even and -1 for odd L4 S4 Simplifying result Given a set of linearly-independent vectors We can find a set of anti-commuting vectors such that These vectors all anti-commute Symmetric matrix Define The magnitude of the product is also correct L4 S5 Decomposing products Make repeated use of Define the inner product of a vector and a bivector L4 S6 General result Grade r-1 Over-check means this term is missing Define the inner product of a vector Remaining term is the outer product and a grade-r term Can prove this is the same as earlier definition of the outer product L4 S7 General product Extend dot and wedge symbols for homogenous multivectors The definition of the outer product is consistent with the earlier definition (requires some proof). This version allows a quick proof of associativity: L4 S8 Reverse, scalar product and commutator The reverse, sometimes written with a dagger Useful sequence Write the scalar product as Occasionally use the commutator product Useful property is that the commutator Scalar product is symmetric with a bivector B preserves grade L4 S9 Rotations Combination of rotations Suppose we now rotate a blade So the product rotor is So the blade rotates as Rotors form a group Fermions? Take a rotated vector through a further rotation The rotor transformation law is Now take the rotor on an excursion through 360 degrees. -
Multiplying and Factoring Matrices
Multiplying and Factoring Matrices The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Strang, Gilbert. “Multiplying and Factoring Matrices.” The American mathematical monthly, vol. 125, no. 3, 2020, pp. 223-230 © 2020 The Author As Published 10.1080/00029890.2018.1408378 Publisher Informa UK Limited Version Author's final manuscript Citable link https://hdl.handle.net/1721.1/126212 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ Multiplying and Factoring Matrices Gilbert Strang MIT [email protected] I believe that the right way to understand matrix multiplication is columns times rows : T b1 . T T AB = a1 ... an . = a1b1 + ··· + anbn . (1) bT n T Each column ak of an m by n matrix multiplies a row of an n by p matrix. The product akbk is an m by p matrix of rank one. The sum of those rank one matrices is AB. T T All columns of akbk are multiples of ak, all rows are multiples of bk . The i, j entry of this rank one matrix is aikbkj . The sum over k produces the i, j entry of AB in “the old way.” Computing with numbers, I still find AB by rows times columns (inner products)! The central ideas of matrix analysis are perfectly expressed as matrix factorizations: A = LU A = QR S = QΛQT A = XΛY T A = UΣV T The last three, with eigenvalues in Λ and singular values in Σ, are often seen as column-row multiplications (a sum of outer products).