TY - JOUR

T1 - Asymptotic cones of embedded singular spaces

AU - Sun, Xiang

AU - Morvan, Jean-Marie

N1 - KAUST 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.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/668620

UR - http://www.intlpress.com/site/pub/pages/journals/items/gic/content/vols/0002/0001/a003/

U2 - 10.4310/gic.2015.v2.n1.a3

DO - 10.4310/gic.2015.v2.n1.a3

M3 - Article

VL - 2

SP - 47

EP - 76

JO - Geometry, Imaging and Computing

JF - Geometry, Imaging and Computing

SN - 2328-8876

IS - 1

ER -