The Geometric Algebra Java Package: Novel Structure Implementation of 5D Geometric Algebra R_ for Object Oriented Euclidean Geometry, Space-Time Physics and Object Oriented Computer Algebra

This paper first briefly reviews the algebraic background of the conformal (homogeneous) model of Euclidean space in Clifford geometric algebra R_<4,1> = Cl(4,1), concentrating on the subalgebra structure. The subalgebras include space-time algebra (STA), Dirac and Pauli algebras, as well as real and complex quaternion algebras, etc. The concept of the Horosphere is introduced along with the definition of subspaces that intuitively correspond to three dimensional Euclidean geometric objects. Algebraic expressions for the motions of these objects and their set theoretic operations are given. It is shown how 3D Euclidean information on positions, orientations and radii can be extracted. The second main part of the paper concentrates on the GeometricAlgebra Java package implementation of the Clifford geometric algebra R_<4,1> = Cl(4,1) and the homogeneous model of 3D Euclidean space. Details are exemplified by looking at the structure and code of the basic MultiVector class and of the 3D Euclidean object model class Sphere. Finally code optimization issues and the ongoing open source project implementation are discussed.