  Using orthogonal projections, we investigate distance of a given elasticity
tensor to classes of elasticity tensors exhibiting particular material
symmetries. These projections depend on the orientation of the elasticity
tensor, hence the distance is obtained as the minimization of corresponding
expressions with respect to the action of the orthogonal group. These
expressions are stated in terms of the eigenvalues of both the given tensor and
the projected one. The process of minimization is facilitated by the fact that,
as we prove, the traces of the corresponding Voigt and dilatation tensors are
invariant under these orthogonal projections. For isotropy, cubic symmetry and
transverse isotropy, we formulate algorithms to find both the orientation and
the eigenvalues of the elasticity tensor that is endowed with a particular
symmetry and is closest to the given elasticity tensor.
