Approximating Surfaces in $R^3$ by Meshes with Guaranteed Regularity

Abstract

We study the problem of approximating a surface $F$ in $R^3$ by a high quality mesh, a piecewise-flat triangulated surface whose triangles are as close as possible to equilateral. The MidNormal algorithm generates a triangular mesh that is guaranteed to have angles in the interval $[49.1^o, 81.8^o]$. As the mesh size $e\rightarrow 0$, the mesh converges pointwise to $F$ through surfaces that are isotopic to $F$. The GradNormal algorithm gives a piecewise-$C^1$ approximation of $F$, with angles in the interval $[35.2^o, 101.5^o]$ as $e\rightarrow 0$. Previously achieved angle bounds were in the interval $[30^o, 120^o]$.