Tutorial on Reflections in Geometric Algebra
in K. Tachibana (ed.) Lecture notes of the International Workshop for "Computational Science with Geometric Algebra" (FCSGA2007), Nagoya University, Japan, 14-21 Feb. 2007, pp. 34-44 (2007).
Abstract: This tutorial focuses on describing the implementation and use of reflections in the geometric algebras of three-dimensional (3D) Euclidean space and in the five-dimensional (5D) conformal model of Euclidean space. In the latter reflections at parallel planes serve to implement translations as well. Combinations of reflections allow to implement all isometric transformations. As a concrete example we treat the symmetries of (2D and 3D) space lattice crystal cells. All 32 point groups of three dimensional crystal cells (10 point groups in 2D) are exclusively described by vectors (two for each cell in 2D, three for one particular cell in 3D) taken from the physical cell. Geometric multiplication of these vectors completely generates all symmetries, including reflections, rotations, inversions, rotaryreflections and rotary-inversions. The inclusion of translations with the help of the 5D conformal model of 3D Euclidean space allows the full formulation of the 230 crystallographic space groups in geometric algebra. The sets of vectors necessary are illustrated in drawings and all symmetry group elements are listed explicitly as geometric vector products. Finally a new free interactive software tool is introduced, that visualizes all symmetry transformations in the way described in the main geometrical part of this tutorial.
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