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 E3, 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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 47-76 |
| Number of pages | 30 |
| Journal | Geometry, Imaging and Computing |
| Volume | 2 |
| Issue number | 1 |
| DOIs | |
| State | Published - 2015 |
Bibliographical note
KAUST Repository Item: Exported on 2021-12-15Acknowledgements: We thank Fran¸cois Golse and Simon Masnou for highlighting interesting results in measure theory that have been useful in our context, and Helmut Pottmann for his help and judicious remarks on a first version of the text.