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 normal moveout (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. Although modern multiparameter NMO descriptions usually are based on the same set of parameters, we show through reparameterization that they can be divided into the same two types of approximations: one assuming an effective medium, the other describing the optical analogue in a medium of constant near-surface velocity. 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. Through a combined use, interesting new applications are suggested -e.g., in diffraction separation or surface-related multiple elimination..