T-Hulls, Discretization, and Partially Ordered Sets in Molecular Shape Analysis

T-Hulls are a generalization of Alpha-Hulls, where the reference to shape analysis is not a single curvature value, but the complete shape of a reference object T. Whereas T-Hulls provide a generalization of convexity theorems and are versatile tools for molecular shape analysis and shape analysis in general for objects in arbitrary dimensions, they are also tools for discretization of shape description as well as for transformations between continuous and discrete models with the aid of a single continuous parameter. This continuous parameter generates an ordering of shapes and if a finite number of such transformations are considered, for example, with reference to a combination of several molecular properties, then a partial ordering is obtained. This special property can be exploited in the molecular context if our goal is the establishment of rigorous connections between quantum chemical and discrete mathematical (such as graph theoretical) models used for molecular characterization in applied fields, such as QSAR or pharmaceutical drug design.