Exploratory orientation data analysis with ω sections

Since the domain of crystallographic orientations is three-dimensional and spherical, insightful visualization of them or visualization of related probability density functions requires (i) exploitation of the effect of a given orientation on the crystallographic axes, (ii) consideration of spherical means of the orientation probability density function, in particular with respect to one-dimensional totally geodesic submanifolds, and (iii) application of projections from the two-dimensional unit sphere S-2 subset of IR3 onto the unit disk D subset of IR2. The familiar crystallographic 'pole figures' are actually mean values of the spherical Radon R-1 transform. The mathematical Radon R-1 transform associates a real-valued function f defined on a sphere with its mean values R(1)f along one-dimensional circles with centre O, the origin of the coordinate system, and spanned by two unit vectors. The family of views suggested here defines omega sections in terms of simultaneous orientational relationships of two different crystal axes with two different specimen directions, such that their superposition yields a user-specified pole probability density function. Thus, the spherical averaging and the spherical projection onto the unit disk determine the distortion of the display. Commonly, spherical projections preserving either volume or angle are favoured. This rich family displays f completely, i.e. if f is given or can be determined unambiguously, then it is uniquely represented by several subsets of these views. A computer code enables the user to specify and control interactively the display of linked views, which is comprehensible as the user is in control of the display.