Abstract

This paper expands the role of the new field of computational topology by surveying methods for incorporating connectedness in shape modeling. Two geometric representations in particular, recurrent models and implicit surfaces, can (often unpredictably) become connected or disconnected based on typical changes in modeling parameters. Two methodologies for controlling connectedness are identified: connectedness loci and Morse theory. The survey concludes by identifying several open problems remaining in shape modeling for computational topology to solve. 1 Introduction One might ask "what is topology?" and receive the proper definition: "the study of open sets." While this answer may enlighten some, the novice puzzles "what is an open set?" at about the same time the definition continues on to add that open sets are whatever a topology defines them to be (so long as they pass a few conditions regarding closure under union and intersection, etc.). This might be the main reason students ...

Citations