Maths for (Games) Programmers
Section 5 - Multivector Methods

    Introduction

Multivectors
    Notations and Coordinates
          Blades     Specific subcomponents     Zeroes
    Inverse frames
          Inverse Frame Units     Extended inverse frames
    The Geometric Product
          Â₂     Â_1.1     Â₃
    Pseudoscalars
          Duality
    Matrix representations
          Â_p.q.r in Â_2^N×2^N     Â₂ in Â_2×2     Â_1.1 in Â_2×2     Â₃ in C_2×2     Â_3.1 in Â_4×4     Â_4.1 in C_4×4     Other matrix representations     Adding Blades

Multivector Products
    Restricted products
          The outer product     The "thin" outer product     The contractive inner product     The semi-commutative inner product     The "fatdot" inner product     The forced Euclidean contractive inner product     Commutator product     AntiCommutator product     Scalar product     Scalar-Pseudoscalar product     Inversive product     Delta products     Conjugative Products     Pure Product Rule     The Intersective Product     Precedence Conventions

Multivector Operations
    Lifts
    Conjugations
          Identity     Negation     Reverse Conjugation     Involution Conjugation     Clifford Conjugation     Mitian Conjugation     Hermitian Conjugation     Modulatory Conjugation     Extension Conjugations     Third bit Conjugation     Dualed Conjuations     Conjugation tabulations     Possible Implimentation     Directation
    Scalar Measures and Normalising
          Unitisation     Magnitude     Conjugated Normalisation     Modulus     Selfscale     Scalar-Normalisation     Maximal-Coordinate-Normalisation     Trace     Determinant     Oppositioning
    Inverses and Integer Powers
          Inverse     Conjugated Roots
    Exponentials and Logarithms
          Integer Powers     Exponential     Exterior Exponential     Logarithm     Hyperbolic Functions     Central Powers     Complex Numbers     Hyperbolic Numbers     Nullic Numbers     Bi-imaginary numbers     Computing Exponentials and Logarithms
    Projections and Perpendiculars
          Projection     Rejection     Projection via anticommution     Scaled Projections     Normalised Projections     Orthogonal Frames
    Intersections and Unions
          Join     Meet     Union     Disjoint     Plunge     Null Blades

Multivector Programming
          Introduction     Existing Multivector Software
    Representing Multivectors
          Small N     Large N     mv C++ wrapper class     Versors and Other Factorisations
    Programing Multivector Products
          Geometric and Restricted Products     Commutator Products     Addition     C multivector product routine     C++ product wrapper     Dual
    Programing Meet and Join
    Multivector Programming Issues
          Whittling     Multivector Inspection     The Delta Product     Null basis vectors     Performance Issues     Coding Issues     Precedence Issues
    MV 1.3
          MV 1.3 Files

Multivectors as Geometric Objects
    Introduction
    Subspaces
          Lines and Planes     Simplexes     Frames
    Higher Dimensional Embeddings
    Homogeneous coordinates
          k-planes
    Affine Model
    Generalised Homogeneous Coordinates
          e₀ and e_¥     Geometric Interpretation Overview     The horosphere point embedding     Dropping to the horosphere     Hyperspheres     k-planes     k-spheres     Interpreting blades as k-spheres     k-antispheres     Example Geometric Manipulations     Pencils     Geometric Interpretation of GHC Blades
    Nonflat Embeddings
    k-conics
          Regeneralised Homogeneous Coordinates     Line Segments
    The "One Up" Embeddings
          Moving off the horosphere     k-planes and k-spheres     k-Planes and k-Spheres
    Spherical Conformal Coordinates
    Tspherical Conformal Coordinates
    Soft Geometry

Multivectors as Transformations
    Bivector Transform
    Lorentz Transforms
    Higher Dimensional Embeddings
           Homogeneous Coordinates     Affine Model     Generalised Homogeneous Coordinates
    Translations
          Affine Model     Generalised Homogeneous Coordinates
    Reflections
          Null Reflection Rule     Affine Model     Generalised Homogeneous Coordinates
    Shears and Strains
    Rotations
          3D Rotations     4D Rotations     Affine Model     Generalised Homogeneous Coordinates
    Inversion
          Transversion
    Dilation and Involution
    Summary of GHC Transformations
    Spinor Transforms
          Example (l(b+e_¥d+ge_¥0))^↑_-1     Implementation via idempotent forms

Multivector Shapes
    k-tubes
    Transforming k-blades
    Transforming k-spheres
          The Slide     The Inside Slide     The Inside Swing     Mutating k-spheres     Transforming using control points

Multivector Mathematical Arcana
    Algebraic Equivalences
          Relativity of Signatures and Grades     Algebraic Product Formulations     Complexification
    Minimal Geometric Algebras
    Bivectors
          Expressed as sum of commuting 2-blades
    Spinors
          Spinor Adjustment Rule     Alternate representations of spinors     Riemann Sphere Representation

Multivector Kinematics
    Rigid Body Kinematics
    Conclusion

References/Source Material for Multivector Methods


Glossary   Contents   Author
Copyright (c) Ian C G Bell 1998
Web Source: www.iancgbell.clara.net/maths or www.bigfoot.com/~iancgbell/maths
Latest Edit: 01 Jun 2007.