Tutorial : Consistent Representations of and Conversions Between 3 D Rotations

In materials science the orientation of a crystal lattice is described by means of a rotation relative to an external reference frame. A number of rotation representations are in use, including Euler angles, rotation matrices, unit quaternions, Rodrigues-Frank vectors and homochoric vectors. Each representation has distinct advantages and disadvantages with respect to the ease of use for calculations and data visualization. It is therefore convenient to be able to easily convert from one representation to another. However, historically, each representation has been implemented using a set of often tacit conventions; separate research groups would implement different sets of conventions, thereby making the comparison of methods and results difficult and confusing. This tutorial article aims to resolve these ambiguities and provide a consistent set of conventions and conversions between common rotational representations, complete with worked examples and a discussion of the trade-offs necessary to resolve all ambiguities. Additionally, an open source Fortran-90 library of conversion routines for the different representations is made available to the community. Submitted to: Modelling Simulation Mater. Sci. Eng. ‡ Corresponding author: Department of Materials Science and Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213-3980, USA; Phone: (412) 268-8527, Fax: (412) 268-7596.