enter Models and representations of classical groups. Clifford algebras, chain geometries over Clifford algebras. Kinematic mappings for Pin and Spin groups. Cayley-Klein geometries. After revising known representations of the group of Euclidean displacements Daniel Klawitter gives a comprehensive introduction into Clifford algebras.
The goal was to demonstrate how the framework of geometric algebra Clifford algebra could unify and illuminate diverse fields of science and engineering. The articles reveal [a] range [of] fields: from quantum physics to robotics, from crystallographic groups to image understanding, from relativistic mechanics to signal processing. Despite this diversity, the combination of these subjects was not felt to be artificial. This book should be July , The present book contains the papers of this scientific meeting and reflects the constantly growing interest in searching the applications of geometric algebra or Clifford algebra in various fields of science.
Geometric algebra includes a lot of techniques from several mathematical theories linear algebra, vector calculus, projective geometry, complex analysis and offers new directions in some unexpected domains like quantum physics, robotics, crystallographic groups, image understanding, relativistic mechanics, signal processing. The volume begins with a preface written by the Editors and a useful list with contributors There are four sections: Algebra and Geometry In conclusion, a very useful book both for beginners and specialists!
Select Interface Language. One research project deals with applications of geometric algebras in computer science. The perception-action cycle PAC as the frame for autonomous behavior relates perception and action in a purposive manner.
The implementation of artificial PACs demands on the fusion of signal theory, computer vision, robotics and neural computing. In this research work we use an interpretation of the Clifford algebra called geometric algebra. Clifford or geometric algebras are well known to pure mathematicans.
The elements in geometric algebras are called multivectors which can be multiplied together using a geometric product. Euclidean, projective and conformal geometry find in geometric algebra the frame where they can reconcile and express their potential. This opens a new alternative for the mathematical treatment of the stratification of the 3D visual space.
Since the Kiel GA applications group set up theoretic bases for dealing with tasks of signal processing , projective geometry , robot kinematics and geometric neural computing. Since the work is extended to practical applications and numerical experiences with respect to different research topics. The most expressive one we use so-far is the conformal geometric algebra. It provides a homogeneous model for stereographically projected points on a hypersphere and therefore couples kinematics with projective geometry. The geometric idea behing this algebra are stereographic projections: Simply speaking, a stereographic projection is one way to generate a flat map of the earth.
The rule for a stereographic projection has a nice geometric description: Think of the earth as transparent sphere, intersected on the equator by an equatorial plane.
Now imagine a light bulb at the north pole n, which shines through the sphere. Each point on the sphere casts a shadow on the paper and that is where it is drawn on the map. Using a homogeneous model for stereographic projected points leads to a cone in space.
Embedding this model in a Clifford algebra leads to the conformal geometric algebra CGA. It is suited to describe conformal geometry, it contains spheres as entities and the conformal transformations as geometric manipulations. Several other research groups deal with Clifford algebras mainly mathematicans and physicists.
Please check out the following links to the GA-community:. In Proc. In