A note on the triangle-centered quadratic interpolation discretization of the shape operator

In this note we consider a simple shape operator discretization for general meshes, based on computing an interpolating quadratic function passing through vertices of a triangle and its edge-adjacent neighbors. This approximation is computationally simple and consistent for a broad class of meshes. However, its convergence properties in the context of mesh optimization problems are not as good as some of the previously proposed techniques and it suffers from instabilities for certain point configurations. In [GGRZ06], we explored the behavior of a number of discretizations of the shape operator on general meshes. Other than the midedge normal discrete operator introduced in that paper, all these operators were using vertex degrees of freedom (DOFs) only, and were obtained one of three minimal stencils: edge-centered (vertices of two triangles sharing an edge), face-centered (vertices of all triangles adjacent to a given triangle), and vertex-centered (vertices of all triangles sharing a given vertex). The stencils are shown in Figure 1. Figure 1: Minimal stencils: edge-centered, triangle-centered and vertex-centered. Once a stencil is fixed, there are two common approaches to defining a shape operator or mean curvature vector: interpolating or approximating quadratic approximation to the surface, and elementary hinge operator averaging. The latter approach is motivated by discrete geometric ideas, i.e., defining shape operators using properties not requiring surface smoothness (see [CSM03]). In this case, the shape operator associated with an edge is approximated by an operator with one principal curvature direction aligned with the edge, and the only non-zero principal curvature magnitude proportional to the angle between triangle normals. In the case of the simplest stencil (two triangles sharing an edge) the two approaches are effectively the same, as only a cylinder can be used to interpolate four points. One type of quadratic fit [Tau95] and discrete hinge averaging [PP93] were considered for vertex-centered stencils in [GGRZ06]. However, [GGRZ06] presents the results only the hinge-averaging operator for the triangle-centered stencil. Quadratic fit can be also applied on the triangle-centered stencil: the number of DOFs for triangles with no vertices of valence 3, exactly matches the number of DOFs needed for a general quadratic function, so a quadratic fit yields an interpolating quadratic function. As it was pointed out in [Zor05] this is sufficient for consistency of discretization, but is in general insufficient for convergence. This note presents the results that were obtained for this operator at the time [GGRZ06] was written, but were omitted from the experimental results due to space limitations and the lack of observed advantages of this operator compared to other formulations. Additionally, we present explicit formulas for the operator obtained using elementary geometry, rather than solving a linear system of equations.