Local Conditions for Triangulating Submanifolds of Euclidean Space

Jean-Daniel Boissonnat (1, 2) , Ramsay Dyer (1, 2) , Arijit Ghosh (3) , Andre Lieutier (4) , Mathijs Wintraecken (5)

1 DATASHAPE - Understanding the Shape of Data
2 CRISAM - Inria Sophia Antipolis - Méditerranée
3 Indian Statistical Institute [Kolkata]
4 Dassault Systèmes
5 IST Austria - Institute of Science and Technology [Klosterneuburg, Austria]

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.

Keywords: Triangulation criteria; Pseudo-manifolds; Submanifolds of Euclidean space; Whitney

Domains: Computer Science [cs]; Computational Geometry [cs.CG]; Computer Science [cs]; Data Structures and Algorithms [cs.DS]; Mathematics [math]; Algebraic Topology [math.AT]; Mathematics [math]; Geometric Topology [math.GT]

Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh, Andre Lieutier, Mathijs Wintraecken. Local Conditions for Triangulating Submanifolds of Euclidean Space. Discrete and Computational Geometry, 2021, 66 (2), pp.666-686. ⟨10.1007/s00454-020-00233-9⟩. ⟨hal-03372073⟩