Triangulating Smooth Submanifolds with Light Scaffolding

We propose an algorithm to sample and mesh a k-submanifold $${\mathcal{M}}$$ of positive reach embedded in $${\mathbb{R}^{d}}$$ . The algorithm first constructs a crude sample of $${\mathcal{M}}$$ . It then refines the sample according to a prescribed parameter $${\varepsilon}$$ , and builds a mesh that approximates $${\mathcal{M}}$$ . Differently from most algorithms that have been developed for meshing surfaces of $${\mathbb{R} ^3}$$ , the refinement phase does not rely on a subdivision of $${\mathbb{R} ^d}$$ (such as a grid or a triangulation of the sample points) since the size of such scaffoldings depends exponentially on the ambient dimension d. Instead, we only compute local stars consisting of k-dimensional simplices around each sample point. By refining the sample, we can ensure that all stars become coherent leading to a k-dimensional triangulated manifold $${\hat{\mathcal{M}}}$$ . The algorithm uses only simple numerical operations. We show that the size of the sample is $${O(\varepsilon ^{-k})}$$ and that $${\hat{\mathcal{M}}}$$ is a good triangulation of $${\mathcal{M}}$$ . More specifically, we show that $${\mathcal{M}}$$ and $${\hat{\mathcal{M}}}$$ are isotopic, that their Hausdorff distance is $${O(\varepsilon ^{2})}$$ and that the maximum angle between their tangent bundles is $${O(\varepsilon )}$$ . The asymptotic complexity of the algorithm is $${T(\varepsilon) = O(\varepsilon ^{-k^2-k})}$$ (for fixed $${\mathcal{M}, d}$$ and k).