Modelling natural objects with α-complexes

The shape of natural objects can be so complicated that only a sampling point set can accurately represent them. Analytic descriptions are too complicated or impossible. Natural objects can be vague and rough with many holes. For this kind of modelling, α-complexes offer advantages over triangulations and hulls at only little extra cost. Geometric and topological descriptions are well formalised, with the flexibility to capture holes and separations. Careful layout of the input sampling point set and the attachment of weights make ‘special modelling effects’ possible. Our approach assumes modelling-by-example: start with a sampling point set from a physical example and take the modelling from there. We explore in this paper the merits of geometric modelling with α-complexes, with emphasis on practical values. We discuss the α-complex as a model description and as a representation scheme. We also show how to run FEM computations on α-complexes.