  One of the greatest theoretical challenges in the build-up to the era of
second-generation gravitational-wave detectors is the modeling of generic
binary waveforms. We introduce an approximation that has the potential to
significantly simplify this problem. We show that generic precessing-binary
inspiral waveforms (covering a seven-dimensional space of intrinsic parameters)
can be mapped to a two-dimensional space of non-precessing binaries,
characterized by the mass ratio and a single effective total spin. The mapping
consists of a time-dependent rotation of the waveforms into the
quadrupole-aligned frame, and is extremely accurate (matches $> 0.99$ with
parameter biases in the total spin of $\Delta \chi \leq 0.04$), even in the
case of transitional precession. In addition, we demonstrate a simple method to
construct hybrid post-Newtonian--numerical-relativity precessing-binary
waveforms in the quadrupole-aligned frame, and provide evidence that our
approximate mapping can be used all the way to the merger. Finally, based on
these results, we outline a general proposal for the construction of generic
waveform models, which will be the focus of future work.
