Polynomial 3D Biharmonic Coordinates and Their Derivatives for Polygonal Cages

Biharmonic coordinates have become a powerful tool for cage-based deformation, owing to their inherent interpolation properties. However, their derivation for polynomial cages in 3D has remained unsolved. To address this, we propose closed-form expressions for polynomial 3D biharmonic coordinates and their derivatives when deformed from polygonal cages using the high-order boundary element method. Our primary contribution lies in the analytical derivation of the kernel integration using recursive differentiation techniques. Due to the enriched deformation space of biharmonic coordinates and the flexibility of polynomial cages, our method supports a broad range of deformations, as demonstrated through extensive experiments.