解説：特集 コンピューテーショナル・インテリジェンスの新展開 —クリフォード代数表現など高次元表現を中心として— Introduction to Clifford’s Geometric Algebra Eckhard HITZER＊ ＊Department of Applied Physics, University of Fukui, Fukui, Japan ＊E-mail: [email protected] Key Words：hypercomplex algebra, hypercomplex analysis, geometry, science, engineering. JL 0004/12/5104–0338 C 2012 SICE erty 15), 21) that any isometry4 from the vector space V Abstract into an inner-product algebra5 A over the field6 K can Geometric algebra was initiated by W.K. Clifford over be uniquely extended to an isometry7 from the Clifford 130 years ago. It unifies all branches of physics, and algebra Cl(V )intoA. The Clifford algebra Cl(V )is has found rich applications in robotics, signal process- the unique associative and multilinear algebra with this ing, ray tracing, virtual reality, computer vision, vec- property. Thus if we wish to generalize methods from tor field processing, tracking, geographic information algebra, analysis, calculus, differential geometry (etc.) systems and neural computing. This tutorial explains of real numbers, complex numbers, and quaternion al- the basics of geometric algebra, with concrete exam- gebra to vector spaces and multivector spaces (which ples of the plane, of 3D space, of spacetime, and the include additional elements representing 2D up to nD popular conformal model. Geometric algebras are ideal subspaces, i.e. plane elements up to hypervolume ele- to represent geometric transformations in the general ments), the study of Clifford algebras becomes unavoid- framework of Clifford groups (also called versor or Lip- able. Indeed repeatedly and independently a long list of schitz groups). Geometric (algebra based) calculus al- Clifford algebras, their subalgebras and in Clifford al- lows e.g., to optimize learning algorithms of Clifford gebras embedded algebras (like octonions 17))ofmany neurons, etc. spaces have been studied and applied historically, often . under different names. 1 Introduction Some of these algebras are complex numbers (and W.K. Clifford (1845-1879), a young English Goldsmid the complex number plane), hyperbolic numbers (split professor of applied mathematics at the University Col- complex numbers, real tessarines), dual numbers, lege of London, published in 1878 in the American Jour- quaternions, biquaternions (complex quaternions), dual nal of Mathematics Pure and Applied a nine page long quaternions, Pl¨ucker coordinates, bicomplex numbers paper on Applications of Grassmann’s Extensive Alge- (commutative quaternions, tessarines, Segre quater- bra. In this paper, the young genius Clifford, stand- nions), Pauli algebra (space algebra), Dirac algebra ing on the shoulders of two giants of algebra: W.R. (space-time algebra, Minkowski algebra), algebra of Hamilton (1805-1865), the inventor of quaternions,and physical space, para-vector algebra, spinor algebra, Lie H.G. Grassmann (1809-1877), the inventor of extensive algebras, Cartan algebra, versor algebra, rotor algebra, algebra, added the measurement of length and angle motor algebra, Clifford bracket algebra, conformal alge- to Grassmann’s abstract and coordinate free algebraic bra, algebra of differential forms, etc. methods for computing with a space and all its sub- Section 1.1 can also be skipped by readers less inter- spaces. Clifford thus unified and generalized in his geo- ested in mathematical definitions. Then follow sections metric algebras (=Clifford algebras) the works of Hamil- on the geometric algebras of the plane Cl(2, 0), of 3D ton and Grassmann by finalizing the fundamental con- space Cl(3, 0), both with many notions also of impor- cept of directed numbers 23). Any Clifford algebra Cl(V ) is generated from an 4A K-isometry between two inner-product spaces is a K-linear 1 inner-product vector space (V , a · b : a, b ∈ V → R) mapping preserving the inner products. 5 by Clifford’s geometric product setting2 the geometric A K-algebra is a K-vector space equipped with an associative and multilinear product. An inner-product K-algebra is a K- product3 of any vector with itself equal to their inner algebra equipped with an inner product structure when taken · product: aa = a a. We indeed have the universal prop- as K-vector space. 6Important fields are real R and complex numbers C,etc. 7 1The inner product defines the measurement of length and angle. That is a K-linear homomorphism preserving the inner prod- 2This setting amounts to an algebra generating relationship. ucts, i.e., a K-linear mapping preserving both the products of 3No product sign will be introduced, simple juxtaposition im- the algebras when taken as rings, and the inner products of the plies the geometric product just like 2x =2× x. algebras when taken as inner-product vector spaces. 338 計測と制御 第 51 巻第4 号 2012 年 4 月号 tance in higher dimensions, and of spacetime Cl(1, 3), pendent definition: aa = a · a, ∀ a ∈ Rp,q,r 7).This conformal geometric algebra Cl(4, 1), and finally on even applies to Clifford analysis (Geometric Calculus). Clifford analysis. For further study we recommend Clifford algebra is thus ideal for computing with geo- e.g. 5). metrical invariants 15). A review of five different ways 1.1 Definitions to define GA, including one definition based on vector Definition of an algebra 25):LetA be a vector space space basis element multiplication rules, and one defi- over the reals R with an additional binary operation nition focusing on Cl(p, q, r)asauniversal associative from A×A to A, denoted here by ◦ (x ◦ y is the algebra, is given in chapter 14 of the textbook 17). product of any x, y ∈A). Then A is an algebra over In general Ak denotes the grade k part of A ∈ R if the following identities hold ∀ x, y, z ∈A,and Cl(p, q, r). The parts of grade 0, (s − k), (k − s), and “scalars” α, β ∈ R : (1,2) Left and right distributivity: (s + k), respectively, of the geometric product of a k- (x + y) ◦ z = x ◦ z + y ◦ z, x ◦ (y + z)=x ◦ y + x ◦ z.(3) vector Ak ∈ Cl(p, q, r)withans-vector Bs ∈ Cl(p, q, r) Compatibility with scalars: (αx) ◦ (βy)=(αβ)(x ◦ y). This means that x ◦ y is bilinear. The binary operation Ak ∗Bs := AkBs0,Ak Bs := AkBss−k, (2) is often referred to as multiplication in A, which is not AkBs := AkBsk−s,Ak ∧ Bs := AkBsk+s, necessarily associative. Definition of inner product space 1): An inner prod- are called scalar product, left contraction (zero for s< uct space is a vector space V over R together with k), right contraction (zero for k<s), and (associative) an inner product map ., . : V × V → R,thatsat- outer product, respectively. These definitions extend by isfies ∀ x, y, z ∈ V and ∀ α ∈ R: (1) Symmetry: linearity to the corresponding products of general multi- x, y = y,x. (2) Linearity in the first argument: vectors. The various derived products of (2) are related, αx, y = αx, y, x + y,z = x, z + y,z. e.g. by Note: We do not assume positive definiteness. Definition of inner product algebra: An inner product (A ∧ B) C = A (B C), algebra, is an algebra equipped with an inner product ∀A, B, C ∈ Cl(p, q, r). (3) A×A→R. Definition of Clifford’s geometric algebra (GA) 6), 17): Note that for vectors a, b in Rp,q,r ⊂ Cl(p, q, r)wehave Let {e1,e2,...,ep,ep+1,...,ep+q,ep+q+1,...,en},with 2 ∧ n = p+q+r, ek = εk, εk =+1fork =1,...,p, εk = −1 ab = a b + a b, for k = p+1,...,p+q, εk =0fork = p+q+1,...,n,be a b = ab = a · b = a ∗ b, (4) an orthonormal base of the inner product vector space p,q,r R with a geometric product according to the multi- where a · b is the usual inner product of Rp,q,r. plication rules For r = 0 we often denote Rp,q = Rp,q,0,and Cl(p, q)=Cl(p, q, 0). For Euclidean vector spaces ekel + elek =2εkδk,l,k,l=1,...n, (1) (n = p)weuseRn = Rn,0 = Rn,0,0,andCl(n)= Cl(n, 0) = Cl(n, 0, 0). The even grade subalgebra of where δk,l is the Kronecker symbol with δk,l =1fork = Cl(p, q, r) is denoted by Cl+(p, q, r), the k-vectors of its l,andδk,l =0fork = l. This non-commutative product basis have only even grades k.Everyk-vector B that and the additional axiom of associativity generate the n can be written as the outer product B = b1 ∧b2 ∧...∧bk 2 -dimensional Clifford geometric algebra Cl(p, q, r)= p,q,r p,q,r of k vectors b1,b2,...,bk ∈ R is called a simple k- Cl(R )=Clp,q,r = Gp,q,r = Rp,q,r over R.The vector or blade. set {eA : A ⊆{1,...,n}} with eA = eh eh ...eh , 1 2 k Definition of outermorphism 7), 18):Anoutermor- 1 ≤ h1 < ... < hk ≤ n, e∅ =1,formsagraded phism is the unique extension to Cl(V ) of a vector (blade) basis of Cl(p, q, r). The grades k range from 0 space map for all a ∈ V , f : a → f(a) ∈ V ,and for scalars, 1 for vectors, 2 for bivectors, s for s-vectors, p,q,r is given by the mapping B = b1 ∧ b2 ∧ ... ∧ bk → up to n for pseudoscalars. The vector space R is in- f(b1) ∧ f(b2) ∧ ... ∧ f(bk)inCl(V ), for every blade cluded in Cl(p, q, r) as the subset of 1-vectors. The gen- B in Cl(V ). eral elements of Cl(p, q, r) are real linear combinations of basis blades eA, called Clifford numbers, multivectors 2.