We propose three independent methods for determining geometrical shape and topological type of local atomic environment in crystals that is represented as a coordination figure. The methods are based on comparison of the edge graphs of the simplified Voronoi polyhedra, maximizing volume of intersection of polyhedra, and comparison of angular fingerprints of polyhedra. We implemented the methods to the program package ToposPro and tested them on 9642 crystal structures of five-coordinated copper and zinc complexes. The methods showed good selectivity and coincidence. We discuss correlations of the coordination figure type with other structure descriptors and perspectives to use the obtained results in knowledge databases.

The article reviews the most important results in the field of numerical mathematics obtained by Professor V. V. Strygin’s disciples and the disciples of his disciples in 2000–2019 in the city of Samara. We present the results on spline interpolation of functions with a boundary layer, methods and algorithms for a posteriori adaptation of computational grids for singularly perturbed boundary value problems and the results on applied wavelet analysis based on spline wavelets.

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Three universal algorithms for geometrical comparison of abstract sets of n points in the Euclidean space R3 are proposed. It is proved that at an accuracy ϵ the efficiency of all the algorithms does not exceed O(n3/ϵ3/2). The most effective algorithm combines the known Hungarian and Kabsch algorithms, but is free of their deficiencies and fast enough to match hundreds of points. The algorithm is applied to compare both finite (ligands) and periodic (nets) chemical objects.

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