Algorithms for target transformations of lattice basis vectors

Simple algorithms are proposed for the transformation of lattice basis vectors to a specific target. In the first case, one of the new basis vectors is aligned to a predefined lattice direction, while in the second case, two of the new basis vectors are brought to a lattice plane with predefined Miller indices. The multi-dimensional generalization of the algorithm is available in the supporting materials. The algorithms are useful for such crystallographic operations as simulation of zone planes (i.e. geometry of electron diffraction patterns) or transformation of a unit cell for surface or cleavage energy calculations. The most general multi-dimensional version of the algorithm may be useful for the analysis of quasiperiodic crystals or as an alternative method of calculating Bézout coefficients. The algorithms are demonstrated both graphically and numerically.