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Geometric Modelling with a-Complexes

The shape of real objects can be so complicated, that only a sampling data point set can accurately represent them. Analytic descriptions are too complicated or impossible. Natural objects, for example, can be vague and rough with many holes. For this kind of modelling, a-complexes offer advantages over triangulations and hulls at little extra computational cost. Geometric and topological descriptions are well-formalised, with the flexibility to capture holes, up to a complete separation. Spatial distribution of the point set and the attachment of weights make ''special modelling effects" possible. We explore in this paper the merits of geometric modelling with a-complexes, with the objective of evaluating their practical value. We discuss the a-complex as a model description and as a representation scheme. Varying the a-value is intuitive, but weighting can be tedious. We present a few strategies. We also show how to run FEM computations on a-complexes. a-Complexes form a useful addition to existing approaches and are applicable to a number of problems not (easily) handled by existing approaches.