Normal cone approximation and offset shape isotopy

Abstract

This work addresses the problem of the approximation of the normals of the offsets of general compact sets in Euclidean spaces. It is proven that for general sampling conditions, it is possible to approximate the gradient vector field of the distance to general compact sets. These conditions involve the μ-reach of the compact set, a recently introduced notion of feature size. As a consequence, we provide a sampling condition that is sufficient to ensure the correctness up to isotopy of a reconstruction given by an offset of the sampling. We also provide a notion of normal cone to general compact sets that is stable under perturbation.