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# Geometry, Imaging and Computing

## Volume 2 (2015)

### Number 1

### Asymptotic cones of embedded singular spaces

Pages: 47 – 76

DOI: http://dx.doi.org/10.4310/GIC.2015.v2.n1.a3

#### Authors

#### Abstract

We use geometric measure theory to introduce the notion of *asymptotic cones* associated with a singular subspace of a Riemannian manifold. This extends the classical notion of *asymptotic directions* usually defined on smooth submanifolds. We get a simple expression of these cones for polyhedra in $\mathbb{E}^3$, as well as convergence and approximation theorems. In particular, if a sequence of singular spaces tends to a smooth submanifold, the corresponding sequence of asymptotic cones tends to the asymptotic cone of the smooth one for a suitable distance function. Moreover, we apply these results to approximate the asymptotic lines of a smooth surface when the surface is approximated by a triangulation.