Many applications in science and engineering deal with three dimensional shapes that move, deform, and/or evolve with time. These applications need computational methods to simulate such shapes in motion for visualization, inspection, prototyping, and further developments. We propose to focus on the problem of inferring topology and geometry of a dynamic shape from an appropriate representation. We argue that the user can be given a choice of maintaining a data structure of appropriate complexity depending on the goal of the simulation. A lighter data structure can be used if the goal is to capture only topology whereas a more complex data structure can be chosen for capturing both geometry and topology. This view point generates a plethora of mathematical and algorithmic questions that we propose to investigate in this project.
Topology and geometry inference of shapes in motion with theoretical guarantee is a difficult but important problem. A key challenge is to keep the update costs for the maintained data structures low. Recent developments in topological analysis of different types of complexes in the context of surface reconstruction and data analysis have opened up the possibility of representing a shape at different levels of complexity depending on the need. An efficient use of these representations in a kinetic setting is
