| We give a definition of the Delaunay triangulation of a point set in a closed Euclidean d-manifold, i.e.\ a compact quotient space of the Euclidean space for a discrete group of isometries (a so-called Bieberbach group or crystallographic group). We describe a geometric criterion to check whether a partition of the manifold actually forms a triangulation (which subsumes that it is a simplicial complex). We provide an algorithm to compute the Delaunay triangulation of the manifold defined by a given set of points, if it exists. Otherwise, the algorithm returns the Delaunay triangulation of a finitely sheeted covering space of the manifold. Whereas there was prior work for the special case of the flat torus, as far as we know this is the first result for general closed Euclidean d-manifolds. This research is motivated by application fields, like computational structural biology for instance, showing a need to perform simulations in quotient spaces of the Euclidean space by more general groups of isometries than the groups generated by d independent translations. |
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