The β-shape and β-complex for three-dimensional spheres

초록

In recent years, there have been extensive studies on biological systems such as proteins. Being one of the most important aspects, the geometry has been more important since the morphology of a molecular system is known to determine the critical functions of the molecule. In the study of the shape and the structure of a molecule, the representation of proximity information among atoms in the molecule is the most fundamental research issue. In this paper, we present a β-shape and a β-complexfor a set of atoms with arbitrary sizes for a faster response to the topological queries among atoms. These concepts are the generalizations of the well-known α-shape and α-complex (and their weighted counterparts as well). To compute a β-shape, we first compute the Voronoi diagram of atoms and then transform the Voronoi diagram to a quasitriangulation which is the topological dual of the Voronoi diagram. Then, we compute a β-complex from the quasitriangulation by analyzing the valid intervals for each simplex in the quasi-triangulation. It is shown that a β-complex can be computed in O(m) time in the worst case from the Voronoi diagram of atoms, where m is the number of simplices in the quasi-triangulation. Then, a β-shape for a particular β consisting of k simplices can be located O(log m + k) time in the worst case from the simplicies in the β-complex sorted according to the interval values.