Maths for
(Games)
Programmers
Section 4 - Multivector Methods
   
Introduction
Multivectors
   
Notations and Coordinates
         
Blades
   
Specific subcomponents
   
Zeroes
   
Inverse frames
   
The Geometric Product
         
Â
2
   
Â
1.1
   
Â
3
   
Overview
   
Pseudoscalars
         
Duality
   
Centrality
   
Bivectors
   
Matrix representations
         
Â
p.q.r
in
Â
2
N
×2
N
   
Â
2
in
Â
2×2
   
Â
1.1
in
Â
2×2
   
Â
3
in
C
2×2
   
Â
3.1
in
Â
4×4
   
Â
4.1
in
C
4×4
   
Â
7
@
Â
5.2
@
C
6
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
         
e
0
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
+
g
e
¥
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
.