Eigen Space of Mesh Distortion Energy Hessian

Mesh distortion optimization is a popular research topic and has wide range of applications in computer graphics, including geometry modeling, variational shape interpolation, UV parameterization, elastoplastic simulation, etc. In recent years, many solvers have been proposed to solve this nonlinear optimization efficiently, among which projected Newton has been shown to have best convergence rate and work well in both 2D and 3D applications. Traditional Newton approach suffers from ill conditioning and indefiniteness of local energy approximation. A crucial step in projected Newton is to fix this issue by projecting energy Hessian onto symmetric positive definite (SPD) cone so as to guarantee the search direction always pointing to decrease the energy locally. Such step relies on time consuming eigen decomposition of element Hessian, which has been addressed by several work before on how to obtain a conjugacy that is as diagonal as possible. In this report, we demonstrate an analytic form of Hessian eigen system for distortion energy defined using principal stretches, which is the most general representation. Compared with existing projected Newton diagonalization approaches, our formulation is more general as it doesn't require the energy to be representable by tensor invariants. In this report, we will only show the derivation for 3D and the extension to 2D case is straightforward.