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.