  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.
