  We present a new method for extracting the instantaneous orbital axis only
from gravitational wave strains of precessing binary systems observed from a
particular observer direction. This method enables us to reconstruct the
co-precessing frame waveforms only from observed quantities for the ideal case
that the signal-to-noise ratio is high enough to analyze the waveforms
directly. Specifically, we do not assume knowledge of the time evolution of the
instantaneous orbital axis and the co-precessing waveforms before analyzing the
data in our method. We test and measure the accuracy of our method using the
numerical relativity simulation data of precessing binary black holes taken
from the SXS Catalog. We show that the direction of the orbital axis is
extracted within $\approx0.02~{\rm rad}$ error from gravitational waves emitted
during the inspiral phase. The co-precessing waveforms are also reconstructed
with high accuracy; the mismatch (assuming white noise) between them and the
original co-precessing waveforms is typically a few times $10^{-3}$ including
the merger-ringdown phase, and can be improved by an order of magnitude
focusing only on the inspiral waveform. In this method, the co-precessing frame
waveforms are not only the purely technical tools for understanding the complex
nature of precessing waveforms but also direct observables.
