Local Conditions for Triangulating Submanifolds of Euclidean Space

Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh, André Lieutier, Mathijs Wintraecken

Abstract: We consider the following setting: suppose that we are given a manifold M in $${\mathbb {R}}^d$$ R d with positive reach. Moreover assume that we have an embedded simplical complex $${\mathcal {A}}$$ A without boundary, whose vertex set lies on the manifold, is sufficiently dense and such that all simplices in $${\mathcal {A}}$$ A have sufficient quality. We prove that if, locally, interiors of the projection of the simplices onto the tangent space do not intersect, then $${\mathcal {A}}$$ A is a triangulation of the manifold, that is, they are homeomorphic.