Multi-scale Geometric Modeling of Ambiguous Shapes with Toleranced Balls and Compoundly Weighted alpha-shapes
Abstract
First, we introduce toleranced balls to model ambiguous geometric objects. A toleranced ball consists of two concentric balls, and interpolating between their radii provides a way to explore a range of possible geometries. We propose to model an ambiguous shape by a collection of toleranced balls, and show that the aforementioned radius interpolation is tantamount to the growth process associated with an additively-multiplicatively weighted Voronoi diagram (also called compoundly weighted or CW). Second and third, we investigate properties of the CW diagram and the associated CW -complex, which provides a ltration called the -complex. Fourth, we sketch a naive algorithm to compute the CW VD. Finally, we use the -complex to assess the quality of models of large protein assemblies, as these models inherently feature ambiguities.