It is possible to derive a general formula for con-stant-velocity, two-dimensional dip moveout (DMO). This serves to unify the many published forms of DMO. Known results for common-offset and shot profile DMO are special cases of the general formula. The analysis is based on the dip-corrected NMO equation, and thus is a kinematic DMO theory.Using a logarithmic stretch of the time axis, efficient fast Fourier transform (FFT) common-offset DMO algorithms can be derived. In the published versions, log variables are introduced into the NMO equation. It can be shown that the resulting impulse response departs from the DMO ellipse, which means that some dips have been improperly handled. A new log-stretch formulation (exact log DMO) which preserves the DMO ellipse can be derived by starting with the analytic impulse response, rather than the NMO equation.Tests on field data indicate that exact log DMO handles shallow, steeply dipping events in agreement with Hale's DMO, while the other log-stretch algorithms degrade these events. In situations where constant-velocity DMO is viable, exact log DMO can be used to create a Hale-quality image at FFT speed.