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 φ-mean-φ−1φ−1 rule that can be implemented by different φ maps. The present work also shows that the involved maps may be computed in closed form by low-degree polynomial expressions.