  We modeled the HeII 4686 line profiles observed in the eta Carinae binary
system close to the 2003.5 spectroscopic event, assuming that they were formed
in the shocked gas that flows at both sides of the contact surface formed by
wind-wind collision. We used a constant flow velocity and added turbulence in
the form of a gaussian velocity distribution. We allowed emission from both the
primary and secondary shocks but introduced infinite opacity at the contact
surface, implying that only the side of the contact cone visible to the
observer contributed to the line profile. Using the orbital parameters of the
binary system derived from the 7 mm light curve during the last spectroscopic
event (Paper II) we were able to reproduce the line profiles obtained with the
HST at different epochs, as well as the line mean velocities obtained with
ground based telescopes. A very important feature of our model is that the line
profile depends on the inclination of the orbital plane; we found that to
explain the latitude dependent mean velocity of the line, scattered into the
line of sight by the Homunculus, the orbit inclination should be close to 90
degrees, meaning that it does not lie in the Homunculus equatorial plane, as
usually assumed. This inclination, together with the relative position of the
stars during the spectroscopic events, allowed us to explain most of the
observational features, like the variation of the Purple Haze with the orbital
phase, and to conciliate the X-ray absorption with the postulated shell effect
used to explain the optical and UV light curves.
