  We present a dynamical model of the high mass X-ray binary LMC X-1 based on
high-resolution optical spectroscopy and extensive optical and near-infrared
photometry. From our new optical data we find an orbital period of P=3.90917
+/- 0.00005 days. We present a refined analysis of the All Sky Monitor data
from RXTE and find an X-ray period of P=3.9094 +/- 0.0008 days, which is
consistent with the optical period. A simple model of Thomson scattering in the
stellar wind can account for the modulation seen in the X-ray light curves. The
V-K color of the star (1.17 +/- 0.05) implies A_V = 2.28 +/- 0.06, which is
much larger than previously assumed. For the secondary star, we measure a
radius of R_2 = 17.0 +/- 0.8 solar radii and a projected rotational velocity of
V_rot*sin(i) = 129.9 +/- 2.2 km/s. Using these measured properties to constrain
the dynamical model, we find an inclination of i = 36.38 +/- 1.92 deg, a
secondary star mass of M_2 = 31.79 +/- 3.48 solar masses, and a black hole mass
of 10.91 +/- 1.41 solar masses. The present location of the secondary star in a
temperature-luminosity diagram is consistent with that of a star with an
initial mass of 35 solar masses that is 5 Myr past the zero-age main sequence.
The star nearly fills its Roche lobe (~90% or more), and owing to the rapid
change in radius with time in its present evolutionary state, it will encounter
its Roche lobe and begin rapid and possibly unstable mass transfer on a
timescale of a few hundred thousand years.
