DOCSLIB.ORG

Inverse and Determinant in 0 to 5 Dimensional Clifford Algebra

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. alent, so that one is left with terms of Clifford products of Inverses of Clifford numbers in 3–dimensions unique ei’s. Table (I) lists the additional r-basis elements and higher have been computed by using matrix for each subsequent dimension after the zero-dimensional 2 representations , since self-contained Clifford algebra G0 scalar element, here given the label e0 with the under- formulations for these inverses have been missing. This standing that e0 = 1. Counting the basis in the Table (I), paper presents general inverse expressions in Clifford a Clifford algebra of dimension d has 2d unique r-basis algebras of up to 5–dimensions. elements. A general Clifford algebra element is referred to as a Clifford number or a multivector, and can be written as a 1 A. Clifford Algebra Basics sum of the r-basis elements from Table (I) with real (com- plex) coefficients. From this representation, a Clifford The Clifford algebra of d-dimensions, Gd, is an ex- number can be separated into a sum of various grades. tended vector algebra over the real (or complex) num- The sum of the components of a specific grade-n is then bers. The algebraic representation can be generated by called an n-vector. The n-vector parts are written as the the orthonormal bases vectors {ei, i =1, ..., d} of the reg- sum of its n-basis components, Xi...ei..., which will be ular vector space Vd. For a vector space with a diagonal referred to as its r-components. It is these r-components metric {g11,g22,...,gdd}, the Clifford product of the ba- and their corresponding products that form the founda- sis 1-vectors satisfy the fundamental product relation: tion of the proofs and discussions to follow. Additionally, one can also distinguish an n-blade as an arXiv:1104.0067v2 [math-ph] 16 Aug 2012 e e = e · e + e ∧ e = g δ + e (1) i j i j i j ij i j ij n-vector for which there exists an alternative basis with ′ Where the inner-product, ei · ej , commutes and matches basis-element ei... in which the n-vector can be written as ′ the standard inner-product of two vectors. The discus- one term xei.... Alternatively, for a given n-vector Xn, if sion here is for a diagonal metric, although the equations there is a single basis element ei... for which ei...Xn results can be generalized to non-diagonal metrics. in a 1-vector, then the n-vector Xn is also an n-blade. The outer-product of two 1-vectors anticommute: This is because, via linear algebra, any combination of ei∧ej = −ej∧ei, and is non-zero for i6=j. The additional 1-vectors is also a 1-blade. Pseudovectors and pseudo- elements of the basis for Gd are generated by the sub- scalars can be referred to as blades, while scalars are not sequent outer-products of the ei basis vectors. As an since scalars have no basis. For all other non-zero grades, example, e12 = e1e2 = e1∧e2 is one of the grade-2 basis an n-vector is generally not an n-blade. elements, or 2-basis. The number of ei in a general ba- For example, in 4–dimensions, the 2-vectors can sis element after reduction in the product is defined as be separated into two different 2-blades: the space 2-blade (xe23 + ye13 + ze12) and the time 2-blade (ue14 + ve24 + we34) used for rotations and boosts in Spe- cial Relativity. Here, the space 2-blade times e123 results a)email: [email protected] in a 1-vector, while the time part requires multiplication Clifford Algebra Inverse and Determinant 2 ple, the 3–dimensional Clifford number, TABLE I. Standard r-Basis Elements dim grade-1 grade-2 grade-3 grade-4 grade-5 #basis X = a0 + a1e1 + a2e2 + a12e12 + (6) 0 e (†) 1 0 a3e3 + a13e13 + a23e23 + a123e123 1 e1 2 2 e2 e12 4 can be written in the complex form, with I = e123, as, 3 e3 e13,e23 e123 8 X = (a0 + Ia123)e0 + (a1 + Ia23)e1 + (7) 4 e4 e14,e24 e124,e134 e1234 16 e34 e234 (a2 − Ia13)e2 + (a12 − Ia3)e12 5 e5 e15,e25 e125,e135 e1235,e1245 e12345 32 In even dimensions, it is important to maintain the or- e35,e45 e145,e235 e1345,e2345 der of the pseudoscalar token I and the basis elements e245,e345 ei..., since the pseudoscalar will anticommute with odd (†)The scalar element, e =1, is grade-zero. 0 grades. The net sign change for commuting a pseudo- scalar with an r-vector is (−1)r(d−1). In odd dimen- by e4 to show it is a 2-blade. In 2 and 3–dimensions, the sions, maintaining product order is not needed since the 2-vectors are pseudoscalars and pseudovectors, respec- pseudoscalar commutes with all grades. The noncommu- tively, and are therefore 2-blades. Blades will be impor- tivity of the even-dimensional pseudoscalar is the main tant in the special-case determinants discussed later. obstacles in constructing even-dimensional complex de- The pseudoscalar e1...d is often given importance in its terminants from those of the previous odd dimension. analogy to the imaginary ı of the Complex numbers. For Another important difference between even and odd this reason, it is often given the label I, although its self- dimensions is in the middle grades. For even dimen- product is not necessarily (−1), but instead given by the sions, there is a middle grade “d/2” which is split between self-product the real and imaginary parts, such as the 1-vector part (a1 − Ia2) of Equation (5). In 4–dimensions, it is the 2 d(d−1)/2 I = e1...de1...d = (−1) g11g22. .gdd (2) space and time bivector parts that become the real and imaginary parts of the complex bivector. This would sug- where it takes d(d−1)/2 anticommutations to reverse the gest that the even-dimensional complex representation is product order in the pseudoscalar from e1...d to ed...1. inherently representation dependent, however, splitting The final value of the pseudoscalar squared depends on the 2-vectors evenly into non-blade half real and half the metric values gii. Clifford algebras usually have a imaginary parts shows that the complex representation diagonal metric with values of +1, −1 and zero. Such a (r,s,t) can be structured as representation independent. Clifford algebra is labeled Gd , where {r,s,t} are the numbers of +1, −1 and zero metric elements respectively. The most common metrics are the Euclidean {1,..., 1} B. The 5–Dimensional Self-Product (d,0,0) for Gd , and the Minkowski metric {1, −1,..., −1} for (1,d−1,0) A general Clifford number in 5–dimensions has 32 com- Gd . The importance of the unit pseudoscalar I in this pa- ponents: one scalar 0-component, five 1-components, ten per is in writing a Clifford number in its complex rep- 2-components, ten 3-components, five 4-components and resentation. In the complex representation, the unit one 5-component, as listed in Table (I). When a general pseudoscalar takes on the role of the imaginary token, al- multivector is multiplied by itself, the contributions to though it does not necessarily square to −1. The complex the grades in the result are based on the sign changes representation is obtained by first multiplying the Clif- resulting from the left and right products of the various ford number by the unit pseudoscalar, a product called r-components. Those products that result in the same the dual or left-dual, dual[X]= IX. To include dimen- sign will commute and contribute, while those that have sions in which I2 = −1, the procedure requires multiply- opposite signs will anticommute and cancel. ing by I4 = 1, with I3 taking X to its third dual.
Load more

Recommended publications
  • 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 .
    [Show full text]
  • Università Degli Studi Di Trieste a Gentle Introduction to Clifford Algebra
    Università degli Studi di Trieste Dipartimento di Fisica Corso di Studi in Fisica Tesi di Laurea Triennale A Gentle Introduction to Clifford Algebra Laureando: Relatore: Daniele Ceravolo prof. Marco Budinich ANNO ACCADEMICO 2015–2016 Contents 1 Introduction 3 1.1 Brief Historical Sketch . 4 2 Heuristic Development of Clifford Algebra 9 2.1 Geometric Product . 9 2.2 Bivectors . 10 2.3 Grading and Blade . 11 2.4 Multivector Algebra . 13 2.4.1 Pseudoscalar and Hodge Duality . 14 2.4.2 Basis and Reciprocal Frames . 14 2.5 Clifford Algebra of the Plane . 15 2.5.1 Relation with Complex Numbers . 16 2.6 Clifford Algebra of Space . 17 2.6.1 Pauli algebra . 18 2.6.2 Relation with Quaternions . 19 2.7 Reflections . 19 2.7.1 Cross Product . 21 2.8 Rotations . 21 2.9 Differentiation . 23 2.9.1 Multivectorial Derivative . 24 2.9.2 Spacetime Derivative . 25 3 Spacetime Algebra 27 3.1 Spacetime Bivectors and Pseudoscalar . 28 3.2 Spacetime Frames . 28 3.3 Relative Vectors . 29 3.4 Even Subalgebra . 29 3.5 Relative Velocity . 30 3.6 Momentum and Wave Vectors . 31 3.7 Lorentz Transformations . 32 3.7.1 Addition of Velocities . 34 1 2 CONTENTS 3.7.2 The Lorentz Group . 34 3.8 Relativistic Visualization . 36 4 Electromagnetism in Clifford Algebra 39 4.1 The Vector Potential . 40 4.2 Electromagnetic Field Strength . 41 4.3 Free Fields . 44 5 Conclusions 47 5.1 Acknowledgements . 48 Chapter 1 Introduction The aim of this thesis is to show how an approach to classical and relativistic physics based on Clifford algebras can shed light on some hidden geometric meanings in our models.
    [Show full text]
  • 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.
    [Show full text]
  • Tutorial on Fourier and Wavelet Transformations in Geometric Algebra
    Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion Tutorial on Fourier and Wavelet Transformations in Geometric Algebra E. Hitzer Department of Applied Physics University of Fukui Japan 17. August 2008, AGACSE 3 Hotel Kloster Nimbschen, Grimma, Leipzig, Germany E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion Acknowledgements I do believe in God’s creative power in having brought it all into being in the first place, I find that studying the natural world is an opportunity to observe the majesty, the elegance, the intricacy of God’s creation. Francis Collins, Director of the US National Human Genome Research Institute, in Time Magazine, 5 Nov. 2006. I thank my wife, my children, my parents. B. Mawardi, G. Sommer, D. Hildenbrand, H. Li, R. Ablamowicz Organizers of AGACSE 3, Leipzig 2008. E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion CONTENTS 1 Geometric Calculus Multivector Functions 2 Clifford GA Fourier Transformations (FT) Definition and properties of GA FTs Applications of GA FTs, discrete and fast versions 3 Quaternion Fourier Transformations (QFT) Basic facts about Quaternions and definition of QFT Properties of QFT and applications 4 Spacetime Fourier Transformation (SFT) GA of spacetime and definition of SFT Applications of SFT 5 Clifford GA Wavelets GA wavelet basics and GA wavelet transformation GA wavelets and uncertainty limits, example of GA Gabor wavelets 6 Conclusion E. Hitzer Department of Applied Physics University of Fukui Japan GA Fourier & Wavelet Transformations Geometric Calculus GA FTs Quaternion FT Spacetime FT GA Wavelets Conclusion Multivector Functions Multivectors, blades, reverse, scalar product Multivector M 2 Gp;q = Clp;q; p + q = n; has k-vector parts (0 ≤ k ≤ n) scalar, vector, bi-vector, .
    [Show full text]
  • Reciprocal Frame Vectors
    Reciprocal Frame Vectors Peeter Joot March 29, 2008 1 Approach without Geometric Algebra. Without employing geometric algebra, one can use the projection operation expressed as a dot product and calculate the a vector orthogonal to a set of other vectors, in the direction of a reference vector. Such a calculation also yields RN results in terms of determinants, and as a side effect produces equations for parallelogram area, parallelopiped volume and higher dimensional analogues as a side effect (without having to employ change of basis diagonalization arguments that don’t work well for higher di- mensional subspaces). 1.1 Orthogonal to one vector The simplest case is the vector perpendicular to another. In anything but R2 there are a whole set of such vectors, so to express this as a non-set result a reference vector is required. Calculation of the coordinate vector for this case follows directly from the dot product. Borrowing the GA term, we subtract the projection to calculate the rejection. Rejuˆ (v) = v − v · uˆ uˆ 1 = (vu2 − v · uu) u2 1 = v e u u − v u u e u2 ∑ i i j j j j i i 1 vi vj = ∑ ujei u2 ui uj 1 ui uj = (u e − u e ) 2 ∑ i j j i v v u i<j i j Thus we can write the rejection of v from uˆ as: 1 1 ui uj ui uj Rej (v) = (1) uˆ 2 ∑ v v e e u i<j i j i j Or introducing some shorthand: uv ui uj Dij = vi vj ue ui uj Dij = ei ej equation 1 can be expressed in a form that will be slightly more convient for larger sets of vectors: 1 ( ) = uv ue Rejuˆ v 2 ∑ Dij Dij (2) u i<j Note that although the GA axiom u2 = u · u has been used in equations 1 and 2 above and the derivation, that was not necessary to prove them.
    [Show full text]
  • Geometric Algebra for Mathematicians
    Geometric Algebra for Mathematicians Kalle Rutanen 09.04.2013 (this is an incomplete draft) 1 Contents 1 Preface 4 1.1 Relation to other texts . 4 1.2 Acknowledgements . 6 2 Preliminaries 11 2.1 Basic definitions . 11 2.2 Vector spaces . 12 2.3 Bilinear spaces . 21 2.4 Finite-dimensional bilinear spaces . 27 2.5 Real bilinear spaces . 29 2.6 Tensor product . 32 2.7 Algebras . 33 2.8 Tensor algebra . 35 2.9 Magnitude . 35 2.10 Topological spaces . 37 2.11 Topological vector spaces . 39 2.12 Norm spaces . 40 2.13 Total derivative . 44 3 Symmetry groups 50 3.1 General linear group . 50 3.2 Scaling linear group . 51 3.3 Orthogonal linear group . 52 3.4 Conformal linear group . 55 3.5 Affine groups . 55 4 Algebraic structure 56 4.1 Clifford algebra . 56 4.2 Morphisms . 58 4.3 Inverse . 59 4.4 Grading . 60 4.5 Exterior product . 64 4.6 Blades . 70 4.7 Scalar product . 72 4.8 Contraction . 74 4.9 Interplay . 80 4.10 Span . 81 4.11 Dual . 87 4.12 Commutator product . 89 4.13 Orthogonality . 90 4.14 Magnitude . 93 4.15 Exponential and trigonometric functions . 94 2 5 Linear transformations 97 5.1 Determinant . 97 5.2 Pinors and unit-versors . 98 5.3 Spinors and rotors . 104 5.4 Projections . 105 5.5 Reflections . 105 5.6 Adjoint . 107 6 Geometric applications 107 6.1 Cramer’s rule . 108 6.2 Gram-Schmidt orthogonalization . 108 6.3 Alternating forms and determinant .
    [Show full text]
  • Young Research Forum on International Conclave On
    Mathematica Eterna Short Communication 2020 Short Communication on Introduction of Exterior algebra Mohammad Adil* Associate Professor, Department of Mathematics, Assiut University, Egypt In mathematics, the exterior product or wedge product of vectors only k-blades, but sums of k-blades; such a calculation is called a k- is an algebraic construction worn in geometry to Vector the k-blades, since they are uncomplicated products of learn areas, volumes, and their advanced dimensional analogues. vectors, are called the easy fundamentals of the algebra. The rank of The external result of two vectors u and v, denoted by u ∧ v, is any k-vector is distinct to be the fewest integer of simple called a bivector and lives in a space called the exterior square, fundamentals of which it is a sum. The exterior product extends to a vector space that is distinctive from the inventive gap of vectors. the complete exterior algebra, so that it makes sense to multiply any The degree of u ∧ v can be interpret as the area of the lozenge two elements of the algebra. Capable of with this product, the with sides u and v, which in three dimensions can also be exterior algebra is an associative algebra, which means that α ∧ compute by means of the cross product of the two vectors. Like (β ∧ γ) = (α ∧ β) ∧ γ for any elements α, β, γ. The k-vectors have the cross product, the exterior product is anticommutative, degree k, meaning that they are sums of products of k vectors. significance that When fundamentals of dissimilar degrees are multiply, the degrees add like multiplication of polynomials.
    [Show full text]
  • Quaternions: a History of Complex Noncommutative Rotation Groups in Theoretical Physics
    QUATERNIONS: A HISTORY OF COMPLEX NONCOMMUTATIVE ROTATION GROUPS IN THEORETICAL PHYSICS by Johannes C. Familton A thesis submitted in partial fulfillment of the requirements for the degree of Ph.D Columbia University 2015 Approved by ______________________________________________________________________ Chairperson of Supervisory Committee _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ Program Authorized to Offer Degree ___________________________________________________________________ Date _______________________________________________________________________________ COLUMBIA UNIVERSITY QUATERNIONS: A HISTORY OF COMPLEX NONCOMMUTATIVE ROTATION GROUPS IN THEORETICAL PHYSICS By Johannes C. Familton Chairperson of the Supervisory Committee: Dr. Bruce Vogeli and Dr Henry O. Pollak Department of Mathematics Education TABLE OF CONTENTS List of Figures......................................................................................................iv List of Tables .......................................................................................................vi Acknowledgements .......................................................................................... vii Chapter I: Introduction ......................................................................................... 1 A. Need for Study ........................................................................................
    [Show full text]
  • A Framework for N-Dimensional Visibility Computations
    A Framework for n-dimensional Visibility Computations L. Aveneau, S. Charneau, L Fuchs and F. Mora Abstract This chapter introduces global visibility computation using Grassmann Algebra. Visibility computation is a fundamental task in computer graphics, as in many other scientific domains. While it is well understood in two dimensions, this does not remain true in higher dimensional spaces. Grassmann Algebra allows to think about visibility at a high level of abstraction, and to design a framework for solving visibility problems in any n-dimensional space, for n ≥ 2. Contrary to Stolfi’s framework which allows only the representa- tion of real lines, it’s algebraic nature dealt naturally without any particular cases. This chapter shows how the space of lines can be defined as a projective space over the bivector vector space. Then line classification, a key point for the visibil- ity computation, is achieved using the exterior product. Actually, line classification turns out to be equivalent to point vs. hyperplane classification relatively to a non- degenerate bilinear form. This ensures well-defined property and computationally robustness. Using the previous result, the lines stabbing a n-dimensional convex face are characterized. This set of lines appears to be the intersection of the decomposable bivectors set (i.e. bivectors that represent a line) and a convex polytope. Moreover, this convex polytope is proved to be minimal. This property allows useful algorith- mic improvements. To illustrate the use of our framework in practice, we present the computation of soft shadows for 3-dimensional illuminated scene. Lilian Aveneau XLIM/SIC, CNRS, University of Poitiers, France, e-mail: [email protected] Sylvain Charneau XLIM/SIC, CNRS, University of Poitiers, France e-mail: [email protected] Laurent Fuchs XLIM/SIC, CNRS, University of Poitiers, France e-mail: [email protected] Fred´ eric´ Mora XLIM/SIC, CNRS, University of Limoges, France e-mail: [email protected] 1 2 L.
    [Show full text]
  • Package 'Clifford'
    Package ‘clifford’ March 8, 2020 Type Package Title Arbitrary Dimensional Clifford Algebras Version 1.0-2 Maintainer Robin K. S. Hankin <[email protected]> Description A suite of routines for Clifford algebras, using the 'Map' class of the Standard Template Library. Canonical reference: Hestenes (1987, ISBN 90-277-1673-0, ``Clifford algebra to geometric calculus''). Special cases including Lorentz transforms, quaternion multiplication, and Grassman algebra, are discussed. Conformal geometric algebra theory is implemented. License GPL (>= 2) LazyData yes Suggests knitr,testthat,onion,lorentz VignetteBuilder knitr Imports Rcpp (>= 0.12.5) LinkingTo Rcpp,BH SystemRequirements C++11 URL https://github.com/RobinHankin/clifford.git BugReports https://github.com/RobinHankin/clifford/issues NeedsCompilation yes Author Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>) Repository CRAN Date/Publication 2020-03-08 20:20:02 UTC R topics documented: clifford-package . .2 allcliff . .4 antivector . .5 1 2 clifford-package as.vector . .6 clifford . .7 Conj.............................................8 drop............................................. 10 dual ............................................. 11 even............................................. 12 Extract.clifford . 12 getcoeffs . 14 grade . 15 homog . 16 lowlevel........................................... 17 magnitude . 18 neg.............................................. 19 numeric_to_clifford . 20 Ops.clifford . 21 print . 24 quaternion . 25 rcliff............................................
    [Show full text]
  • Geometric Algebra: a Computational Framework for Geometrical Applications (Part II: Applications)
    Geometric algebra: a computational framework for geometrical applications (part II: applications) Leo Dorst∗ and Stephen Manny Abstract Geometric algebra is a consistent computational framework in which to de- fine geometric primitives and their relationships. This algebraic approach contains all geometric operators and permits coordinate-free specification of computational constructions. It contains primitives of any dimensionality (rather than just vec- tors). This second paper on the subject uses the basic products to represent ro- tations (naturally incorporating quaternions), intersections, and differentiation. It shows how using well-chosen geometric algebra models, we can eliminate special cases in incidence relationships, yet still have the efficiency of the Plucker¨ coordi- nate intersection computations. Keywords: geometric algebra, Clifford algebra, rotation reprensentation, quaternions, dualization, meet, join, Plucker¨ coordinates, homogeneous coordinates, geometric dif- ferentiation, computational geometry 1 Introduction This is the second of two papers introducing geometric algebra. In the previous paper, we introduced blades, a computational algebraic representation of oriented subspaces, which are the basic element of computation in geometric algebra. We also looked at the geometric product, and two products derived from the geometric product, the inner and outer products. A crucial feature of the geometric product is that it is invertible. From that first paper, you should have gathered that every vector space with an in- ner product has a geometric algebra, whether you choose to use it or not. This paper shows how to call upon this structure for the definition of common geometrical con- structs, ensuring a totally consistent computational framework. The goal is to show ∗Informatics Institute, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands yComputer Science Department, University of Waterloo, Waterloo, Ontario, N2L 3G1, CANADA 0 c 2002 IEEE.
    [Show full text]
  • Geometric Algebra: an Introduction with Applications in Euclidean and Conformal Geometry
    San Jose State University SJSU ScholarWorks Master's Theses Master's Theses and Graduate Research Fall 2013 Geometric Algebra: An Introduction with Applications in Euclidean and Conformal Geometry Richard Alan Miller San Jose State University Follow this and additional works at: https://scholarworks.sjsu.edu/etd_theses Recommended Citation Miller, Richard Alan, "Geometric Algebra: An Introduction with Applications in Euclidean and Conformal Geometry" (2013). Master's Theses. 4396. DOI: https://doi.org/10.31979/etd.chd4-za3f https://scholarworks.sjsu.edu/etd_theses/4396 This Thesis is brought to you for free and open access by the Master's Theses and Graduate Research at SJSU ScholarWorks. It has been accepted for inclusion in Master's Theses by an authorized administrator of SJSU ScholarWorks. For more information, please contact [email protected]. GEOMETRIC ALGEBRA: AN INTRODUCTION WITH APPLICATIONS IN EUCLIDEAN AND CONFORMAL GEOMETRY A Thesis Presented to The Faculty of the Department of Mathematics and Statistics San Jos´eState University In Partial Fulfillment of the Requirements for the Degree Master of Science by Richard A. Miller December 2013 ⃝c 2013 Richard A. Miller ALL RIGHTS RESERVED The Designated Thesis Committee Approves the Thesis Titled GEOMETRIC ALGEBRA: AN INTRODUCTION WITH APPLICATIONS IN EUCLIDEAN AND CONFORMAL GEOMETRY by Richard A. Miller APPROVED FOR THE DEPARTMENT OF MATHEMATICS AND STATISTICS SAN JOSE´ STATE UNIVERSITY December 2013 Dr. Richard Pfiefer Department of Mathematics and Statistics Dr. Richard Kubelka Department of Mathematics and Statistics Dr. Wasin So Department of Mathematics and Statistics ABSTRACT GEOMETRIC ALGEBRA: AN INTRODUCTION WITH APPLICATIONS IN EUCLIDEAN AND CONFORMAL GEOMETRY by Richard A. Miller This thesis presents an introduction to geometric algebra for the uninitiated.
    [Show full text]