Anomalies in quasi-triangulations and beta-complexes of spherical atoms in molecules

The beta-complex is the most compact and efficient representation of molecular structure as it stores the precise proximity among spherical atoms in molecules. Thus, the beta-complex is a powerful tool for solving otherwise difficult shape-related problems in molecular biology. However, to use the beta-complex properly, it is necessary to correctly understand the anomalies of both the quasi-triangulation and the beta-complex. In this paper, we present the details of the anomaly of the beta-complex in relation to the quasi-triangulation. With a proper understanding of anomaly theory, seemingly complicated application problems related to the geometry and topology among spherical balls can be correctly and efficiently solved in rather straightforward computational procedures. We present the theory with examples in both R2R2 and R3R3.

► The beta-complex is the most compact yet precise representation of spherical particles.
► The main difficulty of using the powerful beta-complex lies in an anomaly.
► Interpretations of the anomaly are given for both the beta-complex and the quasi-triangulation.
► If correctly understood, the anomaly facilitates an easier algorithm/solution to previously difficult problems.