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Home , Geometric algebra, Geometric Algebra (book), Rotation (mathematics), Rotor (mathematics), Scalar (mathematics), Spacetime algebra

Geometric (GA)

Werner Benger, 2007 CCT@LSU SciViz

1 Abstract

• (GA) denotes the re-discovery and geometrical interpretation of the applied to real fields. Hereby the so-called „geometrical product“ allows to expand (as used in in 3D) by an invertible to multiply and divide vectors. In two dimenions, the geometric algebra can be interpreted as the algebra of complex . In extends in a natural way into three and corresponds to the well-known there, which are widely used to describe in 3D as an alternative superior to 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 of . 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 (relativity theory), quantum , and last but not least 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: , computer graphics, , 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” • V over K with “ ” • Null-element, One-element, Inverse • Commutative? a b = b a • Associative? (a b) c = a (b c) • ? • 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 (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 in the plane

i sin

cos 5 Historical Roots, II (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 (, 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/ (8 components) • /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 (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 segment or tangent

• Vector-algebra in 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 • (associative): a b  Bivektor – Generalized cross-product from 3D: a b – Not invertible e.g. a b =0 with a≠0, b≠0 but

Multiplikation von Vectoren 11 a b Describes the plane spun by a and b, sign is

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

No unique determination of the generating vectors possible

=

= a b = (a+λb) b b b b =0

Basis-element |a| |b| sin a+λb

Multiplikation von Vektoren 13 Bivectors in R3

• 3 -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, ez ex

Multiplikation von Vektoren 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+(axbz- azbx)exz

Multiplikation von Vektoren 16 Inner product a b • Describes projections

a b = |a| |b| cos = b a

Symmetric (commutative), requires (Metric) as additional structure, not associative (a b) c a (b c)

Multiplikation von Vektoren 17 Comparing the products

• Inner product • Outer product – Not associative – Associative • (a b) c ≠ a (b c) • (a b) c= a (b c) – Commutative – Not commutative • a b = b a • a b ≠ b a – Not invertible – Not invertible – Yields a – 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. Associative: (AB)C = A(BC) 2. Left-distributive: A(B+C) = AB+AC 3. Right-distributive: (B+C)A= BA+CA 4. 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 • (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 “” • Subset of the tensoralgebra • Geometric Product is invertible!

Das Geometrische Produkt 22 Multi-vector components

2 • R : A = A0 + A1 e0 + A2 e1 + A3 e0 e1

2.7819… + • R3: A = A0 + + + +

A1 e0 + A2 e1 + A3 e2 + + + A4 e0 e1+A5 e1 e2+A6 e0 e2 + A e e e 7 0 1 2 +

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 at a 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. Identification with Quaternions 2. Rotation in 2D 3. Rotation in nD 4. 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

 AB=A B=-BA Basiselement

(AB)2 = (AB) (AB) = -(AB) (BA)=-A(BB) A= -1

Rotation 28 Algebra • 2D: complex numbers 2 • i:= exey, i = -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: (complex quaternions, 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 : 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 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

-1 Consider: R v R 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 -1 -1 -1 R v R = R v┴ R + R v║ R -1 R v┴ R =(cos + sin U) v┴ (cos - sin U)

= v┴(cos² - sin² U²) = v┴

-1 U - U -2 U R v R = v┴ + e v║ e = v┴ + v║ e

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 Maxwell Equations Quantum Mechanics Relativity

36 Describing

• Multiple reflections by r1,r2,r3, … are consecutive products of vectors:

– r3r2r1 v r1r2r3 (not possible w. quaternions) • groups in molecules and crystals can be characterized by – three unit vectors a,b,c – 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 operator: μ μ μ μ := e μ with μ= / x , e e = 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 : B = 0 39 Cl3(R) &

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

0 1 0 -i 1 0 = = x ( 1 0 ) 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- (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 (STA) • GA in 4D with Minkowski-Metric (+,-,-,-)

• Chose { 0, 1, 2, 3} 2 2 – where 2 μ ν = μ ν+ ν μ= 2ημν i.e. 0 = - k = 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 “”

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

μ • Four-dimensional gradient := 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 • (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 (, line, plane …) as vectors in 5D • Unions and intersections of objects are algebraic operations (“meet”, “join”) 47 Objects in conform 5D GA

2 Punkt x + e0 + |x| /2 e Paar von Punkten 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 (Second Edition). ISBN 0792355148, Kluwer Academic Publishers (1999)

• Oersted Medal Lecture 2002: Reforming the Mathematical Language of (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 Approach to Gravity (A.N. Lasenby, C.J.L. Doran, Y. Dabrowski, A.D. Challinor, Cavendish Laboratory, Cambridge), astro-ph/9707165

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