Kolmogoroff-Nagumo mean over the affine symplectic group of matrices

The present work shows that Harris' exponential-mean-log averaging rule over the space of optical transference matrices may be regarded as an instance of the Kolmogoroff-Nagumo averaging rule over the affine symplectic group. As such, Harris' averaging rule may be generalized to a phi-mean-phi^{-1} rule that can be implemented by different phi maps. The present work also shows that the involved maps may be computed in closed form by low-degree polynomial expressions.