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)|
|Number of pages||30|
|Journal||Geometry, Imaging and Computing|
|State||Published - 2015|
Bibliographical noteKAUST Repository Item: Exported on 2021-12-15
Acknowledgements: 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.