Geometric Algebra (GA)
Werner Benger, 2007
CCT@LSU SciViz
1
Abstract
•
Geometric Algebra (GA) denotes the re-discovery and geometrical interpretation of
the Clifford algebra applied to real fields. Hereby the so-called „geometrical product“
allows to expand linear algebra (as used in vector calculus in 3D) by an invertible
operation to multiply and divide vectors. In two dimenions, the geometric algebra can
be interpreted as the algebra of complex numbers. In extends in a natural way into
three dimensions and corresponds to the well-known quaternions there, which are
widely used to describe rotations in 3D as an alternative superior to matrix calculus.
However, in contrast to quaternions, GA comes with a direct geometrical
interpretation of the respective operations and allows a much finer differentation
among the involved objects than is achieveable via quaternions. Moreover, the
formalism of GA is independent from the dimension of space. For instance, rotations
and reflections of objects of arbitrary dimensions can be easily described intuitively
and generic in spaces of arbitrary higher dimensions.
•
Due to the elegance of the GA and its wide applicabililty it is sometimes denoted as a
new „fundamental language of mathematics“. Its unified formalism covers domains
such as differential geometry (relativity theory), quantum mechanics, robotics and last
but not least computer graphics in a natural way.
•
This talk will present the basics of Geometric Algebra and specifically emphasizes on
the visualization of its elementary operations. Furthermore, the potential of GA will be
demonstrated via usage in various application domains.
2
Motivation of GA
• Unification of many domains: quantum
mechanics, computer graphics, general relativity,
robotics…
• Completing algebraic operations on vectors
• Unified concept for geometry and algebra
• Superior formalism for rotations in arbitrary
dimensions
• Explicit geometrical interpretation of the
involved objects and operations on them
3
Definition: “Algebra”
• Vector space V over field K with
multiplication “”
• Null-element, One-element, Inverse
• Commutative? ab = ba
• Associative? (ab)c = a(bc)
• Division algebra?
• a≠0  a-1 such that a  a-1 = 1 = a-1  a
• Alternatively: ab=0  a=0 or b=0
4
Historical Roots
• Complex Plane (Gauss ~1800)
•
•
•
•
Real/Imaginary part: a+ib where i2= -1
Associative, commutative division-algebra
Polar representation: r ei = r ( cos  + i sin  )
Multiplication corresponds to rotation in the plane
i
sin 

cos 
5
Historical Roots, II
William Rowan Hamilton (1805-65)
invents Quaternions (1844):
– Generalization of complex numbers:
• 4 components, non commutative: ab  ba
(in general)
• Basic idea: ii=jj=kk= ijk = -1
• Alternative to younger vector- and matrix
algebra (Josiah Willard Gibbs, 1839-1903)
• p=(p,p), q=(q,q), pq=(pq - p  q , pq + pq
+ pq)
• rotation in R3 are around axis of the vector
component: v’ = q v q-1
6
Historical Roots, III
• Construction by Cayley-Dickson
(a,b)(c,d) = (ac-d *b, *a d+cb)
– hypercomplex numbers:
• octaves/octonions (8 components)
• sedenions/hexadekanions (16 components)
• …
– incremental loss of
• commutativity (quaternions,…)
• associativity (octonions,…)
• division algebra (sedenions,…)
7
Renaissance of the GA
1878: Clifford introduces “geometric algebra”, but dies at age 34
 superseded by Gibb’s vector calculus
1920er: Renaissance in quantum mechanics (Pauli, Dirac)
algebra on complex fields
no geometrical interpretation
1966-2005 David Hestenes (Arizona State University) revives the
geometrical interpretation
1997: Gravitation theory using GA (Lasenby, Doran, Gull; Cambridge)
2001: Geometric Algebra at SIGGRAPH (L. Dorst, S. Mann)
8
9
Geometry and Vectors
• Geometric interpretation of a vector
– Directed line segment or tangent
• Vector-algebra in Euclidean Geometry or Tp(M)
• Addition / subtraction of vectors a+b
• Multiplication / division by scalars  a
• Multiplication / Division of vectors??
Multiplication of vectors
10
Complete Vector-algebra?
• Invertible product of vectors?
• What means vector-division “a/b” ?
• ab=C  b=a-1C
• Note: C not necessarily a vector!
• Inner product (not associative): ab  Skalar
– Not invertible
e.g. ab =0 with a≠0, b≠0 but orthogonal
• Outer product (associative): ab  Bivektor
– Generalized cross-product from 3D: ab
– Not invertible
e.g. ab =0 with a≠0, b≠0 but parallel
Multiplikation von Vectoren
11
Bivector ab
Describes the plane spun by a and b,
sign is orientation
ab
ba = -ab
Defined in arbitrary dimensions, anti-symmetric ( not commutative),
associative, distributive, spans a vector space, does not require additional
structures
Multiplikation von Vektoren
12
Constructing Bivectors
No unique determination of the generating vectors possible
=
ab = (a+λb)b
bb =0
=
Basis-element
|a| |b| sin 
Multiplikation von Vektoren
b
a+λb
13
Bivectors in R3
• 3 Basis-elements
exey,
eyez,
ezex
• Generalization: exeyez is a volume
Multiplikation von Vektoren
14
Vectorspace of
Bivectors
Linear combinations possible
e.g.:
exey,
Multiplikation von Vektoren
ezex
15
Coordinate representation
of “”-product in R3
• Generic Bivector:
A = Axy exey + Ayz eyez + Azx ezex
• (axex + ayey + azez)(bxex + byey + bzez)=
axexbxex + axexbyey + axexbzez +
ayeybxex + ayeybyey + ayeybzez +
azezbxex + azezbyey + azezbzez =
(axby - aybx)exy+(aybz-azby)eyz+(axbzazbx)exz
Multiplikation von Vektoren
16
Inner product ab
• Describes projections
ab = |a| |b| cos  = ba
Symmetric (commutative), requires quadratic form (Metric) as additional
structure, not associative (ab)c a(bc)
Multiplikation von Vektoren
17
Comparing the products
• Inner product
– Not associative
• (ab)c ≠ a(bc)
– Commutative
• ab = ba
– Not invertible
– Yields a scalar
• Outer product
– Associative
• (ab)c= a(bc)
– Not commutative
• ab ≠ ba
– Not invertible
– Yields a bivector
18
Geometric Product
1. Requirements and definition
2. Structure of the operands
3. Calculus using GP
4. Rotations using GP
Das Geometrische Produkt
19
Requirements to GP
•
For elements A,B,C of a vector
space with quadratic form Q(v)
[i.e. a metric g(u,v) = Q(u+v) - Q(u) – Q(v)]
we require:
1.
2.
3.
4.
Associative:
(AB)C = A(BC)
Left-distributive: A(B+C) = AB+AC
Right-distributive: (B+C)A= BA+CA
Scalar product: A2 = Q(A) = |A|2
Das Geometrische Produkt
20
Properties of the GP
• Right-angled triangle
|a+b|2 = |a|2+|b|2
(A+B)(A+B) = AA+BA+AB+BB = A2 + B2
AB = -BA for AB = 0 anti-symm if orthogonal
• However: not purely anti-symmetric
|AB|2 =|A|2 |B|2 for AB = 0 (i.e. A,B parallel: B=A)
Das Geometrische Produkt
21
Geometric Product
• William Kingdon Clifford (1845-79):
• Combine inner and outer product to defined
the geometric product AB (1878):
AB := AB  AB
• Result is not a vector, but the sum of a
scalar + bivector!
• Operates on “multivectors”
• Subset of the tensoralgebra
• Geometric Product is invertible!
Das Geometrische Produkt
22
Multi-vector components
• R 2:
A = A0 + A1 e0 + A2 e1 + A3 e0e1
2.7819…
• R 3:
+
A=
A0
+
A1 e0 + A2 e1 +
A3 e2
+
A4 e0 e1+A5 e1 e2+A6 e0 e2
+
A7 e0 e1 e2
+
+
+
+
+
+
Struktur von Multivektoren
23
Structure of Multi-vectors
Linear combination of anti-symmetric basis elements
2n components
0D
1 Scalar
1D
1 Scalar, 1 Vector
2D
1 Scalar, 2 Vectors, 1 Bivector
3D
1 Scalar, 3 Vectors, 3 Bivectors, 1 Volume
4D
1 Scalar, 4 Vectors, 6 Bivectors, 4 Volume, 1 Hyper-volume
5D
…
Struktur von Multivektoren
24
Inversion
• Given vectors a,b:
ab = ½ (ab + ba)
symmetric part
ab = ½ (ab - ba)
anti-symmetric part
ab = -(ab)  (exeyez) Dual in 3D
Rechnen mit Multivektoren
25
Reflection at a Vector
• Unit vector n, arbitrary vector v
Vector v projected to n: v║=(v  n) n
Reflected vector w = v┴ – v║ = v – 2v║
thus w = v – 2(v  n) n
with GP w = v – 2[½(vn+nv) ] n = v – vnn
– nvn

w = -nvn
Rechnen mit Multivektoren
26
Rotations
1.
2.
3.
4.
Identification with Quaternions
Rotation in 2D
Rotation in nD
Rotation of arbitrary Multivectors in
nD
Rotation
27
Geometrical Quadrate
Consider (AB)2 of Bivector-basis element
where |A|=1, |B|=1, AB = 0
Basiselement
 AB=AB=-BA
(AB)2 = (AB) (AB) = -(AB) (BA)=-A(BB) A= -1
Rotation
28
Quaternion Algebra
• 2D: complex numbers
• i:= exey,
i2 = -1
• 3D: quaternions
• i:= exey= exey, j:= eyez = eyez,
k:=exez=exez
• i2 = -1, j2 = -1 , k2 = -1
• ijk = (exey)(eyez)(exez) = -1
• 4D: Biquaternions (complex
quaternions, spacetime algebra)
Rotation
29
Rotation and GA
Right-multiplication of Vectors by Bivectors
ex i = ex (exey) = (exex ) ey= ey
=
ey i = ey(exey)=-ey(eyex)= -ex
=
Rotation
30
Generic Rotation in 2D
• Multiple Rotation
ex i i = (ex i) i = ey i = -ex = -1 ex
• Arbitrary vector
A = Ax ex + Ay ey
A i = Ax ex i + Ay ey i = Ax ey - Ay ex
• Rotation by arbitrary angle:
A cos + A i sin ≡ “A e i”
rotates vector A by angle  in plane i
Inverse rotation: Ai = -iA :   -
 A ei = e-i A
Rotation
31
Rotor in 2D
• Rotor
R := ei = cos + i sin
mit
i² = -1
A ei = e-i A = e-i/2 A ei/2 = R A R-1
With R=e-i/2 “Rotor”
R-1=ei/2 “inverse Rotor”
A R-2 = R2 A = R A R-1
• Product of rotors is multiple rotation
R=ABCD, R-1=DCBA is “reverse” R
Rotation
32
Rotor in nD
• Rotor in plane U, Vektor v:
R = cos + sin U
U² = -1
Expect: Rv or vR-1 or R v R-1
• Problem: With arbitrary vector v there
would be a tri-vector component:
Rv = v cos + sin (U  v + U  v )
iff Uv ≠ 0 ( v not coplanar with U)
Rotation
33
Rotation in nD
Consider: R v R-1 mit v =v┴ + v║ :
– We have: Uv┴ = 0 d.h. Uv┴ =Uv┴
=u1u2v┴= - u1v┴u2= v┴ u1 u2= v┴U =v┴U
i.e. v┴ commutes with U, thus also R
R v R-1 = R v┴ R-1 + R v║ R-1
R v┴ R-1 =(cos + sin U) v┴ (cos - sin U)
= v┴(cos² - sin² U²) = v┴
R v R-1 = v┴ + eU v║ e-U = v┴ + v║ e-2U
Rotation
34
Rotation as multiple reflection
• Alternative Interpretation:
– Reflect vector v by vector n, then by vector m:
• v  - nvn  m nvn m = mn v nm
• Operation mn is Scalar+Bivector (Rotor!)
• Rotor:
R = mn
• Inverse Rotor: R-1 = nm
• Theorem: Rotation is consecutive
reflection on two corresponding vectors
with the rotation angle equal to twice the
angle between these vectors
Rotation
35
Applications
Crystallography
Differential Geometry
Maxwell Equations
Quantum Mechanics
Relativity
36
Describing Symmetries
• Multiple reflections by r1,r2,r3, … are
consecutive products of vectors:
– r 3 r 2 r 1 v r1 r 2 r 3
(not possible w. quaternions)
• Symmetry groups in molecules and
crystals can be characterized by
– three unit vectors a,b,c
– Integer triple {p,q,r}
– where (ab)p = (bc)q = (ca)r = -1
e.g.: Methane (Tetrahedron) {3,3,3},
Benzene {6,2,2}
37
Differential Geometry
Derivative operator:
:= eμ μ with μ=/xμ, eμe=μ
Applicable to arbitrary multi-vectors
E.G.: with v a vector field:
v = v +   v
where v
Gradient (Scalar)
and
v
Curl (Bivector)
38
Maxwell in 3D
– Faraday-Field:
F = E + B
:=exeyez
– Current density:
J =  - j
– Maxwell-Equation: F/ t +  F = J
F = E + B = E + E + B + B
Scalar :
E = 
Vector :
E / t + B = -j
Bivector:
 B / t + E = 0
Pseudoscalar:
B = 0
39
Cl3(R) & Spinors
• GA in 3D can be represented via Pauli-matrices:
0 1
x =
( )
1 0
1 0
0 -i
y =
( )
+i 0
z =
( )
0 -1
• 4 complex numbers  8 components = 23
• Basis-vectors {ex,ey,ez} with GP provide same algebraic
properties as Pauli-matrices {x,y,z}
• Pauli-Spinor  (2 complex numbers, 4 components),
due to  *= real, can be written as
 = ½ eB
thus is a Rotor (even multi-vector: 1 Scalar, 3 bivector-component),
i.e.  is the “operation” to stretch and rotate  describes
interaction (of an elementary particle) with a magnetic field
40
Spacetime Algebra (STA)
• GA in 4D with Minkowski-Metric (+,-,-,-)
• Chose orthogonal Basis {0, 1, 2, 3}
– where 2μν = μν+ νμ= 2ημν i.e. 02 = -k2 = 1
• Structure: 1,4,6,4,1 ( n4 , 16-dimensional )
– Bivector-Basis: k := k 0
– Pseudo-scalar:  0 1 2 3 = 123
1
{μ} {k, k}
{μ}

1 Scalar 4 Vector 6 Bivectors 4 Pseudo-vectors 1 Pseudo-scalar
41
Basis-Bivectors in STA
k: 3 timelike bi-vectors
k : 3 spacelike bivectors
z
x
y
x  yz
42
Structure of Bivectors
Any bi-vector can be written as
–
–
–
–
B = Bkk = ak k + bkk = a + b
a,b: 3-Vectors (relative 0)
a timelike component
b spacelike component
Classification in
– “complex” Bivector:
No common axes, spans the full
4D space
– “simple” Bivector:
One common axis, can be
reduced to a single “Blade”
43
Spacetime-Rotor
• Spacetime-rotor: R = eB =ea+b  e|B| B/|B|
R = ea+b= eaeb =
[cosh a + sinh a ] [ cos b +  sin b ] =
[cosh |a| + a/|a| sinh |a| ] [cos |b| +  b/|b| sin|b| ]
• Interpretation:
rotation in spacelike plane b by angle |b|
hyperbolic rotation in timelike plane a=a 0 with
“boost-factor” (velocity) tanh|a|
 Lorentz-transformation in a , 0 !
44
Maxwell Equations in 4D
• Four-dimensional gradient := μμ
• Elektro-magnetic 4-potential A:
– F = A = A - A
with A=0 is Lorentz-gauge condition
– Faraday-Field: F = (E + B) 0
Pure Bivector (3D:vector + bi-vector), but complex:
E timelike component, B spacelike
• Maxwell-Equation:
F = J
45
Dirac-Equation
• Relativistic Momentum in Schrödingereqn:
– E=p2/2m  E2 = m2 – p2
(α0mc²
+ ∑ αj pj c)  = i ħ  /  t
where αj Dirac-matrices (44)
in Dirac-basis: 0 = α0, i = α0 αi mit [μ,ν] = 2 ημν
covariant formulation
∑ μ μ  = mc² 
• In GA basis vectors {0, 1, 2, 3} provide
same algebraic properties as Dirac
matrices:
  = mc²  0
46
GA in Computergraphics
• Homogeneous Coordinates (4D):
• Additional coordinate e, 3-vector: Ai / A
• Allows unified handling of directions and
locations, standard in OpenGL
• conform, homogeneous coordinates
(5D):
• Additional coordinates e0, e
• Signature (+,+,+,+,-) , e0e=-1, |e0| = |e| =0
• Allows describing geometric objekts (sphere,
line, plane …) as vectors in 5D
• Unions and intersections of objects are
algebraic operations (“meet”, “join”)
47
Objects in conform 5D GA
Punkt
Paar von Punkten
x + e0 + |x|2/2
e
ab
Linie
abe
Kreis
abc
Ebene
ab c e
Kugel
abcd
48
Implementations
• Runtime evaluation
– geoma (2001-2005),
GABLE (symbolic GA)
• Matrix-based
– CLU (2003)
• Code-Generation
– Gaigen (-2005)
• Template Meta
Programming
– GLuCat, BOOST (~2003)
• Extending programming
languages (proposed)
49
Literatur
http://modelingnts.la.asu.edu/
http://www.mrao.cam.ac.uk/˜clifford
•
David Hestenes: New Foundations for Classical Mechanics (Second Edition). ISBN
0792355148, Kluwer Academic Publishers (1999)
•
Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics
(David Hestenes)
•
Geometric (Clifford) Algebra: a practical tool for efficient geometrical representation
(Leo Dorst, University of Amsterdam)
•
An Introduction to the Mathematics of the Space-Time Algebra (Richard E. Harke,
University of Texas)
•
EUROGRAPHICS 2004 Tutorial: Geometric Algebra and its Application to Computer
Graphics (D. Hildenbrand, D. Fontijne, C. Perwass and L. Dorst)
•
Rotating Astrophysical Systems and a Gauge Theory Approach to Gravity (A.N.
Lasenby, C.J.L. Doran, Y. Dabrowski, A.D. Challinor, Cavendish Laboratory,
Cambridge), astro-ph/9707165
50