Triangulation of stratifolds – StratMesh

In the field of computational geometry, we are addressing the complex task of efficiently computing provably-correct triangulations of manifold with singularities, that can be seen as stratifolds. These structures are prevalent in various domains including robotics, control theory and learning from geometric data. Despite significant efforts, there is still a lack of efficient algorithms for this task, particularly in higher dimensions. Our goal is to develop both theory and efficient algorithms for triangulating manifolds with singularities, thereby addressing this gap. This project also aims at reinvigorating the French computational geometry community, which has been shrinking despite its previous global leadership in the field.

Our primary objective is to create provably-correct triangulation algorithms for stratified spaces, which are sets that can be split into smooth pieces or strata. We aim to focus on stratified spaces that are the projection of smooth manifolds, as these are common in many applications such as robotics, control theory and learning from geometric data. To achieve this, we distinguish two cases. In small dimensions, we plan to bypass the computation of a formula describing the projection of the manifold and directly triangulate the singular and smooth parts of the projection. In higher dimensions, we plan to first analyze the types of singularities that appear in specific settings and then design new algorithms to triangulate and connect them correctly. Finally we plan to work on specific applications and to develop software usable by researchers in those application domains.