Green Coordinates for Triquad Cages in 3D

We introduce Green coordinates for triquad cages in 3D. Based on Green’s third identity, Green coordinates allow defining the harmonic deformation of a 3D point inside a cage as a linear combination of its vertices and face normals. Using appropriate Neumann boundary conditions, the resulting deformations are quasi-conformal in 3D, and thus best-preserve the local deformed geometry, in that volumetric conformal 3D deformations do not exist unless rigid. Most coordinate systems use cages made of triangles, yet quads are in general favored by artists as those align naturally onto important geometric features of the 3D shapes, such as the limbs of a character, without introducing arbitrary asymmetric deformations and representation. While triangle cages admit per-face constant normals and result in a single Green normal-coordinate per triangle, the case of quad cages is at the same time more involved (as the normal varies along non-planar quads) and more flexible (as many different mathematical models allow defining the smooth geometry of a quad interpolating its four edges). We consider bilinear quads, and we introduce a new Neumann boundary condition resulting in a simple set of four additional normal-coordinates per quad. Our coordinates remain quasi-conformal in 3D, and we demonstrate their superior behavior under non-trivial deformations of realistic triquad cages.