Analysis of a method to parameterize planar curves immersed in triangulations

Details Analysis of a method to parameterize planar curves immersed in triangulations Analysis of a method to parameterize planar curves immersed in triangulations

2011-09-16

arxiv.org/abs/1109.5890v1

Mathematics

Numerical Analysis

Scientific paper

We prove that a planar C2-regular boundary \Gamma can always be parameterized with its closest point projection \pi over a certain collection of edges \Gamma_h in an ambient triangulation, by making simple assumptions on the background mesh. For \Gamma_h, we select the edges that have both vertices on one side of \Gamma and belong to a triangle that has a vertex on the other side. By assuming a quasi-uniform family of background meshes, a sufficiently small mesh size h and that certain angles in each mesh are acute, we prove that \pi : \Gamma_h \mapsto \Gamma is a homeomorphism and that it is C1 on each edge in \Gamma_h . We provide bounds for the Jacobian of the parameterization and local estimates for the required mesh size, which could be used in adapting the ambient triangulation. Such a parameterization was first proposed in [17] where it was applied to the construction of a high-order immersed boundary method on a class of planar piecewise C2-curves.

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