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.

**References: **

[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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