Py_convrot: rotation conventions, to understand and to apply

Abstract

Rotation is a core crystallographic operation. Two sets of Cartesian coordinates of each point of a rotated object, those before and after rotation, are linearly related, and the coefficients of these linear combinations can be represented in matrix form. This 3 × 3 matrix is unique for all points and thus describes unambiguously a particular rotation. However, its nine elements are mutually dependent and are not interpretable in a straightforward way. To describe rotations by independent and comprehensible parameters, crystallographic software usually refers to Euler or to polar angles. In crystallography and cryo-electron microscopy, there exists a large choice of conventions, making direct comparison of rotation parameters difficult and sometimes confusing. The program py_convrot, written in Python, is a converter of parameters describing rotations. In particular, it deals with all possible choices of polar angles and with all kinds of Euler angles, including all choices of rotation axes and rotation directions. Using a menu, a user can build their own rotation parameterization; its action can be viewed with an interactive graphical tool, Demo. The tables in this article and the extended help pages of the program describe details of these parameterizations and the decomposition of rotation matrices into all types of parameters. The program allows orthogonalization conventions and symmetry operations to be taken into account. This makes the program and its supporting materials both an illustrative teaching material, especially for non-specialists in mathematics and computing, and a tool for practical use.