Approximating shapes with hierarchies and topologies

This is the second paper in the series on shape decompositions and their use as shape approximations. This timewe investigate hierarchical and topological shape decompositions or hierarchies and topologies.We showed earlier that bounded decompositions behave the same way as shapes do. The same holds for hierarchies and topologies, which are special kinds of bounded decompositions. They are distinguished by their algebraic structures, which have many important properties to facilitate their application as shape approximations. We provide an account of their properties with an emphasis on their application.