  Intensity scintillations of radio pulsars are known to originate from
interference between waves scattered by the electron density irregularities of
interstellar plasma, often leading to parabolic arcs in the two-dimensional
power spectrum of the recorded dynamic spectrum. The degree of arc curvature
depends on the distance to the scattering plasma and its transverse velocity
with respect to the line-of-sight. We report the observation of annual and
orbital variations in the curvature of scintillation arcs over a period of 16
years for the bright millisecond pulsar, PSR J0437-4715. These variations are
the signature of the relative transverse motions of the Earth, pulsar, and
scattering medium, which we model to obtain precise measurements of parameters
of the pulsar's binary orbit and the scattering medium itself. We observe two
clear scintillation arcs in most of our $>$5000 observations and we show that
they originate from scattering by thin screens located at distances $D_1 = 89.8
\pm 0.4$ pc and $D_2 = 124 \pm 3$ pc from Earth. The best-fit scattering model
we derive for the brightest arc yields the pulsar's orbital inclination angle
$i = 137.1 \pm 0.3^\circ$, and longitude of ascending node,
$\Omega=206.3\pm0.4^\circ$. Using scintillation arcs for precise astrometry and
orbital dynamics can be superior to modelling variations in the diffractive
scintillation timescale, because the arc curvature is independent of variations
in the level of turbulence of interstellar plasma. This technique can be used
in combination with pulsar timing to determine the full three-dimensional
orbital geometries of binary pulsars, and provides parameters essential for
testing theories of gravity and constraining neutron star masses.
