E. Hitzer, C. Perwass
Algorithm for conversion between geometric algebra versor notation and conventional crystallographic symmetry-operation symbols
preprint, 14 pp. (2009).

Introduction: This paper establishes an algorithm for the conversion of conformal geometric algebra (GA) [3, 4] versor symbols of space group symmetry-operations [6?8, 10] to standard symmetry-operation symbols of crystallography [5]. The algorithm is written in the mathematical language of geometric algebra [2?4], but it takes up basic algorithmic ideas from [1]. The geometric algebra treatment simplifies the algorithm, due to the seamless use of the geometric product for operations like intersection, projection, rejection; and the compact conformal versor notation for all symmetry operations and for geometric elements like lines and planes.
The transformations between the set of three geometric symmetry vectors a,b,c, used for generating multivector versors, and the set of three conventional crystal cell vectors a,b,c of [5] have already been fully specified in [8] complete with origin shift vectors. In order to apply the algorithm described in the present work, all locations, axis vectors and trace vectors must be computed and oriented with respect to the conventional crystall cell, i.e. its origin and its three cell vectors.
Section 2 reviews the notation for symmetry operations used in ITA2005 [5] together with representative sets of examples in tabulated form. Section 3 reviews the conformal geometric algebra description of points, affine points, lines and planes, their direction, location, parallelness, intersection (meet), as well as the projection of translation vectors. Section 4 introduces the notions of coordinate system, reciprocal coordinate vectors, coordinates, axis lines and basal planes. Section 5 shows how to determine the conventional crystallographic location points of lines and planes. Section 6 shows how to determine the crystallographic trace vectors of planes from intersections with basal coordinate planes. Section 7 explains how to obtain the crystallographic positive sense of a vector. Section 8 introduces to the choice of variable symbols t in {x,y,z} for parametrizing lines and planes. Finally Section 9 presents the full conversion algorithm.

[1] K. Stroz, A systematic approach to the derivation of standard orientation-location parts of symmetry-operation symbols, Acta Crystallographica Section A, A63, pp. 447?454, 2007.
[2] Doran, C. and Lasenby, A. (2003). Geometric Algebra for Physicists, CUP, Cambridge UK.
[3] Dorst, L. and Fontijne, D. and Mann, S. (2007). Geometric Algebra for Computer Science: An Object-oriented Approach to Geometry, Morgan Kaufmann Series in Computer Graphics, San Francisco.
[4] Li, H. (2008). Invariant Algebras and Geometric Reasoning, World Scientific, Singapore.
[5] Hahn, T. (2005). Int. Tables for Crystallography, 5th ed., Vol. A, Springer, Dordrecht.
[6] Hestenes, D. (2002). Point groups and space groups in geometric algebra, edited by L. Dorst et al., Applications of Geometric Algebra in Computer Science and Engineering, Birkhauser, Basel, 3?34.
[7] Hestenes, D. and Holt, J. (2007). The Crystallographic Space Groups in Geometric Algebra, JMP, Vol. 48, 023514.
[8] E. Hitzer, C. Perwass, Interactive 3D Space Group Visualization with CLUCalc and the Clifford Geometric Algebra Description of Space Groups, accepted for Proc. 8th Int. Conf. on Clifford Algebras (ICCA8) and their Appl. in Math. Phys., 26-30 May 2008, Las Campinas, Brazil (2008).
[9] Hitzer, E. and Tachibana, K. and S. Buchholz, and Yu, I. (2009) Carrier Method for the General Evaluation and Control of Pose, Molecular Conformation, Tracking, and the Like, Adv. appl. Clifford alg., Online First.
[10] C. Perwass, E. Hitzer, (2005). Space Group Visualizer, free download for non-commercial users: www.spacegroup.info . The Space Group Visualizer can be purchased from Raytrix GmbH, www.raytrix.de .

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