Geometric Algebra GA

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Geometric Algebra (GA)

• Werner Benger, 2007
• CCT_at_LSU SciViz

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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.

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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

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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?b0 ? a0 or b0

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Historical Roots

• Complex Plane (Gauss 1800)
• Real/Imaginary part aib 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 ?
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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 iijjkk 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

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Historical Roots, III

• Construction by Cayley-Dickson
• (a,b)(c,d) (ac-d b, a dcb)
• hypercomplex numbers
• octaves/octonions (8 components)
• sedenions/hexadekanions (16 components)
• incremental loss of
• commutativity (quaternions,)
• associativity (octonions,)
• division algebra (sedenions,)

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Renaissance of the GA

• 1878 Clifford introduces geometric algebra,
  but dies at age 34
• ? superseded by Gibbs 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)

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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 ab
• Multiplication / division by scalars ? a
• Multiplication / Division of vectors??

Multiplication of vectors
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Complete Vector-algebra?

• Invertible product of vectors?
• What means vector-division a/b ?
• abC ? ba-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
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Bivector a?b

• Describes the plane spun by a and b, sign is
  orientation

b?a -a?b
a?b
Defined in arbitrary dimensions, anti-symmetric
(? not commutative), associative, distributive,
spans a vector space, does not require additional
structures
Multiplikation von Vektoren
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Constructing Bivectors

• No unique determination of the generating vectors
  possible

a?b (a?b)?b
b
b?b 0
Basis-element
a?b
a b sin ?
Multiplikation von Vektoren
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Bivectors in R3

• 3 Basis-elements
• ex?ey, ey?ez, ez?ex
• Generalization ex?ey?ez is a volume

Multiplikation von Vektoren
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Vectorspace of Bivectors
Linear combinations possible e.g. ex?ey,
ez?ex
Multiplikation von Vektoren
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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(axbz-azbx)exz

Multiplikation von Vektoren
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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
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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

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Geometric Product

• Requirements and definition
• Structure of the operands
• Calculus using GP
• Rotations using GP

Das Geometrische Produkt
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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(uv) - Q(u) Q(v) we require
• Associative (AB)C A(BC)
• Left-distributive A(BC) ABAC
• Right-distributive (BC)A BACA
• Scalar product A2 Q(A) A2

Das Geometrische Produkt
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Properties of the GP

• Right-angled triangle
• ab2 a2b2
• (AB)(AB) AABAABBB A2 B2
• AB -BA for A?B 0 anti-symm if orthogonal
• However not purely anti-symmetric
• AB2 A2 B2 for A?B 0 (i.e. A,B
  parallel B?A)

Das Geometrische Produkt
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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
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Multi-vector components

• R2 A A0 A1 e0 A2 e1 A3 e0?e1
• R3 A
• A0
• A1 e0 A2 e1 A3 e2
  
• A4 e0 ?e1A5 e1 ?e2A6 e0 ?e2
• A7 e0 ?e1 ?e2

2.7819







Struktur von Multivektoren
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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
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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
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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 2½(vnnv) n v vnn nvn
• ? w -nvn

Rechnen mit Multivektoren
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Rotations

• Identification with Quaternions
• Rotation in 2D
• Rotation in nD
• Rotation of arbitrary Multivectors in nD

Rotation
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Geometrical Quadrate

• Consider (AB)2 of Bivector-basis element where
  A1, B1, A?B 0
• ? ABA?B-BA
• (AB)2 (AB) (AB) -(AB) (BA)-A(BB) A -1

Basiselement
Rotation
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Quaternion Algebra

• 2D complex numbers
• i exey, i2 -1
• 3D quaternions
• i ex?ey exey, j ey?ez eyez,
  kex?ezexez
• i2 -1, j2 -1 , k2 -1
• ijk (exey)(eyez)(exez) -1
• 4D Biquaternions (complex quaternions, spacetime
  algebra)

Rotation
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Rotation and GA

• Right-multiplication of Vectors by Bivectors
• ex i ex (exey) (exex ) ey ey
• 
  
• ey i ey(exey)-ey(eyex) -ex
• 
  

Rotation
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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
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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 Re-i?/2 Rotor
• R-1ei?/2 inverse Rotor
• A R-2 R2 A R A R-1
• Product of rotors is multiple rotation RABCD,
  R-1DCBA is reverse R

Rotation
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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
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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
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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 ScalarBivector (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
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Applications

• Crystallography
• Differential Geometry
• Maxwell Equations
• Quantum Mechanics
• Relativity

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Describing Symmetries

• Multiple reflections by r1,r2,r3, are
  consecutive products of vectors
• r3r2r1 v r1r2r3 (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

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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)

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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

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Cl3(R) Spinors

• GA in 3D can be represented via Pauli-matrices
• 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

??x ( )
??z ( )
??y ( )
0 1 1 0
1 0 0 -1
0 -i i 0
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Spacetime Algebra (STA)

• GA in 4D with Minkowski-Metric (,-,-,-)
• Chose orthogonal Basis ?0, ?1, ?2, ?3
• where 2?µ??? ?µ?? ???µ 2?µ? 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

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Basis-Bivectors in STA

• ?k 3 timelike bi-vectors
• ??k 3 spacelike
  bivectors

?z
?x
?y
??x ? ?y?z
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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

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Spacetime-Rotor

• Spacetime-rotor R eB ea?b ? eB 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
  sinb
• Interpretation
• rotation in spacelike plane b by angle b
• hyperbolic rotation in timelike plane a?a ?0
  with boost-factor (velocity) tanha
• ? Lorentz-transformation in ?a , ?0 !

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Maxwell Equations in 4D

• Four-dimensional gradient ? ?µ?µ
• Elektro-magnetic 4-potential A
• F ??A ?A - ??A
• with ??A0 is Lorentz-gauge condition
• Faraday-Field F (E ?B) ?0
• Pure Bivector (3Dvector bi-vector), but
  complex
• E timelike component, B spacelike
• Maxwell-Equation ?F J

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Dirac-Equation

• Relativistic Momentum in Schrödingereqn
• Ep2/2m ? E2 m2 p2
• (a0mc² ? aj pj c) ? i h ?? / ? t
• where aj Dirac-matrices (4?4)
• in Dirac-basis ?0 a0, ?i a0 ai mit ?µ,??
  2 ?µ?
• covariant formulation
• ? ?µ ?µ ? mc² ?
• In GA basis vectors ?0, ?1, ?2, ?3 provide same
  algebraic properties as Dirac matrices
• ? ? mc² ? ?0

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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)

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Objects in conform 5D GA
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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)

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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