  $\eta$ Carinae is an extremely luminous and energetic colliding-wind binary.
The combination of its orbit and orientation, with respect to our line of
sight, enables direct investigation of the conditions and geometry of the
colliding winds. We analyse optical He I 5876 and 7065 $\unicode{x212B}$ line
profiles from the Global Jet Watch observatories covering the last 1.3 orbital
periods. The sustained coverage throughout apastron reveals the distinct
dynamics of the emitting versus absorbing components: the emission lines follow
orbital velocities whilst one of the absorption lines is detected only around
apastron ($0.08 < \phi < 0.95$) and exhibits velocities that deviate
substantially from the orbital motion. To interpret these deviations, we
conjecture that this He I absorption component is formed in the post-shock
primary wind, and is only detected when our line of sight intersects with the
shock cone formed by the collision of the two winds. We formulate a geometrical
model for the colliding winds in terms of a hyperboloid in which the opening
angle and location of its apex are parameterised in terms of the ratio of the
wind momentum of the primary star to that of companion. We fit this geometrical
model to the absorption velocities, finding results that are concordant with
the panchromatic observations and simulations of $\eta$ Carinae. The model
presented here is an extremely sensitive probe of the exact geometry of the
wind momentum balance of binary stars, and can be extended to probe the
latitudinal dependence of mass loss.
