Algorithm for finding all k-nearest neighbours in three-dimensional scattered points and its application in reverse engineering

A fast and exact algorithm for computing the k-nearest neighbours, or k-closest points in terms of Euclidean distance, for all data in three-dimensional point clouds is presented that avoids using complicated Voronoi diagrams or Dirichlet tessellations. Experimental evidence suggests that the algorithm has a timing of O(n) for most practical values of k under the condition: k < 0.05 n, where n is the number of three-dimensional points in the cloud. Case studies are presented to illustrate the robustness and efficiency of the method and a comparison is made to an existing exact method.

Discrete versus continual description of solid state reaction dynamics from the angle of meaningful simulation

Chemical dynamics provides quite a number of examples of interesting and useful discrete models. But it catches one's eye that the majority of them are from the field of homogeneous chemistry. Whereas the chemical individuality of solid substances is represented in discrete terms of crystal lattices, the conventional description of solid state reaction dynamics is essentially continual. The recent progress in the theory of random mosaics and theory of planigons opens the way for developing an alternative discrete description in terms of Dirichlet tessellations. In the present paper the two approaches are compared from the angle of meaningful simulation. It seems that this may be of interest not only for chemists but also in the broad context of developing and employing discrete dynamical models.