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 ⊂ R. 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 : R → 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 define Phong projection, as in the paper: Definition 1.1. A point p̂ = ξ1v1 + ξ2v2 + ξ3v3 on triangle t with vertices vi is a Phong projection of p ∈ R if: Ψ(ξ1, ξ2, ξ3) (ξ1v1 + ξ2v2 + ξ3v3 − p) = 0, (1) ξ1 + ξ2 + ξ3 = 1, (2) ξi ≥ 0. (3) Definition 1.2. The Phong projection of a point p onto a triangle mesh M is the closest Phong projection with respect to every triangle of M. Note that the Phong projection onto even a single triangle is generally not unique. Consider the affine subspace through v1 orthogonal to T1. All points in that subspace project to v1. If the intersection between this subspace and the analogous one for v2 is not empty (as will generally happen for D ≥ 4), both v1 and v2 will be Phong projections of points in the intersection. In the paper, we provide experimental evidence that for reasonable meshes and points p close to the mesh, this does not happen; when it does, we break ties arbitrarily. Also, unlike Euclidean projection, Phong projection might not exist at all. In Section 3 of this document, we give an outline of how one might prove that for well-tessellated meshes Phong projection is guaranteed to exist for points p close to the mesh. The remainder of this document is organized as follows. In Sections 1.1-1.4 we deal with tangent plane interpolation and define Ψ, first for mesh edges and then for triangle interiors. Section 2 shows that Ψ is continuous under some mild conditions. In Section 3 we give an informal sketch of how to prove that the Phong projection based on Ψ is well-defined. Throughout these sections, we use some simple algebraic results;