Abstract: Let the patch of a partner in a protein complex be the collection of atoms accounting for the interaction. To improve our understanding of the structure-function relationship, we present a patch model decoupling the topological and geometric properties. While the geometry is classically encoded by the atomic positions, the topology is recorded in a graph encoding the relative position of concentric shells partitioning the interface atoms. The topological - geometric duality provides the basis of a generic dynamic programming based algorithm comparing patches at the shell level, which may favor topological or geometric features. On the biological side, we address four questions, using 249 co-crystallized heterodimers organized in biological families. First, we dissect the morphology of binding patches, and show that Nature enjoyed the topological and geometric degrees of freedom independently while retaining a finite set of qualitatively distinct topological signatures. Second, we argue that our shell-based comparison is effective to perform atomic-level comparisons, and show that topological similarity is a less stringent than geometric similarity. We also use the topological versus geometric duality to exhibit topo-rigid patches, whose topology (but not geometry) remains stable upon docking. Third, we use our comparison algorithms to infer specificity related information amidst a database of complexes. Finally, we exhibit a descriptor outperforming its contenders to predict the binding affinities of the affinity benchmark. The softwares developed with this paper are available from http://team.inria.fr/abs/vorpatch_compatch/