Discrete bending forces and their Jacobians

Computation of bending forces on triangle meshes is required for numerous simulation and geometry processing applications. A common quantity in many bending models is the hinge angle between two adjacent triangles. This angle is straightforward to compute, and its gradient with respect to vertex positions (required for the forces) is easily found in the literature. However, its Hessian, which is required for efficient numerics (e.g., implicit time stepping, Newton-based energy minimization) is not documented in the literature. Readily available computations of the Hessian, such as those produced by symbolic algebra systems, or by autodifferentiation codes, are expensive to compute. We present compact, easily reproducible, closed form expressions for the Hessian. Compared to the automatic differentiation, we measure up to 7× speedup for the evaluation of the bending forces and their gradients.