Phong Projection in Higher Dimensions

We supplement our paper Weighted Averages on Surfaces [1] with technical details, additional intuition, and proofs related to Phong projection. We also show how our generalized barycentric coordinates reduce to Moving Least Squares over Euclidean space. The motivation and applications of Phong projection are discussed in the paper [1]. 1 Phong projection Let M = (V, E,F) be a triangle mesh with vertices V ⊂ RD. Each vertex has an associated tangent plane (taken e.g. from the Loop limit surface), represented by two basis vectors, which we assume to be orthonormal. Consider a triangle with vertices v1,v2,v3 and denote the tangent planes at these vertices as T1, T2, T3 ∈ R2×D. Let Ψ(ξ1, ξ2, ξ3) be an interpolated basis for the tangent plane at the point on the triangle with barycentric coordinates ξ1, ξ2, ξ3. The function Ψ: R3 → R2×D must continuously interpolate the tangent planes to the triangle interiors over the entire mesh. Defining Ψ is a non-trivial task (which we will tackle later) because a tangent plane can be specified using different bases, while the interpolant should be independent of this choice of basis and also consistent on edges and vertices shared by multiple triangles. Once we do have such a Ψ, we can defin