In many present-day applications in seismic processing, the assumption of a homogeneous model leads to simple yet powerful approximations, which also work well when heterogeneity is not negligible. While the classical NMO hyperbola assumes an effective constant velocity medium, de Bazelaire, based on optical projections, introduced an alternative way to account for heterogeneity by shifting the reference time rather than the velocity. In this work, we provide new insights into the auxiliary medium concept and introduce a generalized osculating equation, which allows for the forward and inverse transformation between the effective and the optical domain, thereby providing a unified view on currently used stacking approximations. Supported by synthetic tests, we reveal that all higher order operators can be described in and transformed between both domains, which, through combined use, suggests interesting new applications, like, i. e., in diffraction separation or surface-related multiple elimination.