Maths for (Games) Programmers Section 4 - Multivector Methods

Introduction

Multivectors
Notations and Coordinates
Blades     Specific subcomponents     Zeroes     Inverse frames
The Geometric Product
Â2     Â1.1     Â3     Overview
Pseudoscalars
Duality     Centrality
Bivectors
Matrix representations
Âp.q.r in Â2N×2N     Â2 in Â2×2     Â1.1 in Â2×2     Â3 in C2×2     Â3.1 in Â4×4     Â4.1 in C4×4     Â7@Â5.2@C6 in C8×8     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     Trace     The New Norm     Determinant     Oppositioning
Inverses and Integer Powers
Inverse
Exponentials and Logarithms
Introduction     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
Multivectors expressed as summed commuters
Bivectors expressed as sum of commuting 2-blades
Conclusion

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     Contents of k-spheres     k-antispheres     e--negation     e+-negation     Example Geometric Manipulations     Pencils
Point Versors
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
Perspective Projection
Summary of GHC Transformations
Spinor Transforms
Example (l(b+e¥d+ge¥0))-1     Implementation via idempotentised forms

Multivector Shapes
Unions
k-tubes
Superpositions
Transforming k-spheres
The Slide     The Inside Slide     The Inside Swing     Mutating k-spheres     Transforming using control points
e+ seperation

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, 2014
Web Source: www.iancgbell.clara.net/maths
Latest Edit: 28 Oct 2014.