Triangulating Smooth Submanifolds with Light Scaffolding

Abstract

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).