Maths for (Games) Programmers
Section 4 - Multivector Methods

    Introduction

Multivectors
    Notations and Coordinates
          Blades     Specific subcomponents     Zeroes     Inverse frames
    The Geometric Product
          Â₂     Â_1.1     Â₃     Overview
    Pseudoscalars
          Duality     Centrality
    Bivectors
    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     Â₇@Â_5.2@C₆ in C_8×8     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     Rescaled Product     Pure Product Rule     The Intersective Product     Precedence Conventions

Multivector Operations
    Lifts
    Conjugations
          Identity     Negation     Reverse Conjugation     Involution Conjugation     Clifford Conjugation     Mitian Conjugation     Hermitian Conjugation     e-negating 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     Dorst/Valkenburg Normalisation     Trace     Determinant     Oppositioning
    Inverses and Powers
          Inverse     Integer Powers     Square Roots
    Exponentials and Logarithms
          Introduction     Exponential     Exterior Exponential     Logarithm     Hyperbolic Functions     Central Powers     Complex Numbers     Hyperbolic Numbers     Nullic Numbers     Bi-imaginary numbers     Computing Exponentials and Logarithms     Logarithm of bivector exponentiation
    Projections and Perpendiculars
          Projection     Rejection     Projection via anticommution     Scaled Projections     Normalised Projections     Orthogonal Frames
    Intersections and Unions
          Join     Meet     Union     Disjoint     Plunge     Null Blades
    Multivectors expressed as summed commuters
          Bivectors expressed as sum of commuting 2-blades
    Conclusion

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     The Power Distance     Summing Spheres
    k-planes
    k-spheres
          Interpreting blades as k-spheres     Contents of k-spheres     k-antispheres     e_--negation     e₊-negation     Example Geometric Manipulations     Pencils
    Geometric Interpretation of GHC Blades
          Point Versors     Convergent Point Projection
    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
    "Projective" GHC SubAlgebra
    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
    Perspective Projection
    Summary of GHC Transformations
    Spinor Transforms
          Example (l(b+e_¥d+ge_¥0))^↑_-1     Implementation via idempotentised forms

Multivector Mathematical Arcana
    Algebraic Equivalences
          Relativity of Signatures and Grades     Algebraic Product Formulations     Complexification
    Minimal Geometric Algebras
    Spinors
          Spinor Adjustment Rule     Alternate representations of spinors     Riemann Sphere Representation
    Conclusion

References/Source Material for Multivector Methods


Glossary   Contents   Author
Copyright (c) Ian C G Bell 1998, 2019
Web Source: www.iancgbell.clara.net/maths
Latest Edit: 23 Sep 2019.