KAUST DepartmentApplied Mathematics and Computational Science Program
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Permanent link to this recordhttp://hdl.handle.net/10754/668620
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AbstractWe 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.
CitationSun, X., & Morvan, J.-M. (2015). Asymptotic cones of embedded singular spaces. Geometry, Imaging and Computing, 2(1), 47–76. doi:10.4310/gic.2015.v2.n1.a3
SponsorsWe 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.
PublisherInternational Press of Boston
JournalGeometry, Imaging and Computing