research papers 430 https://doi.org/10.1107/S1600576717001157 J. Appl. Cryst. (2017). 50, 430–440 Received 15 November 2016 Accepted 23 January 2017 Edited by Th. Proffen, Oak Ridge National Laboratory, USA Keywords: three-dimensional rotations; Euler angles; three-dimensional visualization; fundamental zones; Rodrigues vectors. Supporting information: this article has supporting information at journals.iucr.org/j Three-dimensional texture visualization approaches: theoretical analysis and examples Patrick G. Callahan, a McLean Echlin, a Tresa M. Pollock, a Saransh Singh b and Marc De Graef b * a Materials Department, UC Santa Barbara, Santa Barbara, CA 93106, USA, and b Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA. *Correspondence e-mail: degraef@cmu.edu Crystallographic textures are commonly represented in terms of Euler angle triplets and contour plots of planar sections through Euler space. In this paper, the basic theory is provided for the creation of alternative orientation representations using three-dimensional visualizations. The use of homochoric, cubochoric, Rodrigues and stereographic orientation representations is discussed, and illustrations are provided of fundamental zones for all rotational point-group symmetries. A connection is made to the more traditional Euler space representations. An extensive set of three-dimensional visualizations in both standard and anaglyph movies is available. 1. Introduction The texture of a polycrystalline material is frequently displayed in the form of pole figures, inverse pole figures and contour plots of the orientation distribution function (ODF) (Kocks et al. , 1998). While these representations are instruc- tive and useful, they also hide the true mathematical nature of orientation or rotation space (we will use the terms rotation and orientation interchangeably in this paper). In the texture community, Euler angles form the dominant rotation repre- sentation, despite their well known shortcomings. Euler angles suffer from various degeneracies, including the fact that the identity orientation (i.e. no rotation at all) can be represented by an infinite number of Euler angle triplets of the form ð’ 1 ; 0; ’ 1 Þ in the Bunge zxz convention. Furthermore, for a given crystal/sample symmetry, the asymmetric unit or fundamental zone in Euler space in general has curved boundaries, which can make visual representations and the presence of symmetry difficult to interpret. The creation of uniform random samples in Euler space inside a given fundamental zone is not always straightforward. Many texture-related computations, for instance the misorientation between two grains, are usually performed using a different orientation representation, since the metric properties of Euler space make such computations difficult. Despite these difficulties, and the fact that Euler space is an infinite periodic space, the Euler angle representation has for many decades held a central position in texture research. In this paper, we examine a few alternative representations in the context of modern three-dimensional visualization tools; such repre- sentations are aimed at augmenting the more traditional sectional ODF visualizations. For background literature on the representation of rotations and textures, we refer the reader to Grimmer (1980), Heinz & Neumann (1991), Mason & Schuh (2009) and Patala et al. (2012). ISSN 1600-5767 # 2017 International Union of Crystallography