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

Xiang Sun (Visual Computing Center, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia)

Jean-Marie Morvan (Visual Computing Center, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia; and Institut Camille Jordan, Université Claude Bernard Lyon 1, Villeurbanne, France)

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.

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