Analysis of a method to parameterize planar curves immersed in triangulations

We prove that a planar $C^2$-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 imposing restrictions on the size of triangles near the curve and by requesting that certain angles in the mesh be strictly acute, we prove that $\pi:\Gamma_h\rightarrow\Gamma$ is a homeomorphism, that it is $C^1$ on each edge in $\Gamma_h$ and provide bounds for the Jacobian of the parameterization. The assumptions on the background mesh are both easy to satisfy in practice and conveniently verified in computer implementations. The parameterization analyzed here was previously proposed by the authors and applied to the construction of high-order curved finite elements on a class of planar piecewise $C^2$-curves.