As a natural extension to harmonic maps, biharmonic maps have been found to outperform them in the context of, e.g., 2D planar deformations. However, 3D biharmonic coordinates and their derivatives have remained unexplored. In this work, we derive closed-form expressions for 3D biharmonic coordinates and their derivatives for triangular cages. These formulas enable precise and numerically stable computation of biharmonic coordinates in 3D, thus filling a missing component in the family of boundary interpolation schemes. Among the potential applications, we demonstrate our extension to triangular cage deformation enhancing existing Green coordinates with increased expressivity, and their usage for variational surface deformation providing an enriched subspace. Moreover, those lead to closed-form expressions for the recent Somigliana coordinates, which are revealed as a composition of Green coordinates and derivatives of our biharmonic coordinates. This relationship facilitates efficient and precise evaluation of Somigliana coordinates, eliminating the need for ineffective quadrature rules.