Eta Carinae is one of the most massive observable binaries. Yet determination of its orbital and physical parameters is hampered by obscuring winds. However the effects of the strong, colliding winds changes with phase due to the high orbital eccentricity. We wanted to improve measures of the orbital parameters and to determine the mechanisms that produce the relatively brief, phase-locked minimum as detected throughout the electromagnetic spectrum. We conducted intense monitoring of the He ii λ4686 line in η Carinae for 10 months in the year 2014, gathering ~300 high S/N spectra with ground- and space-based telescopes. We also used published spectra at the FOS4 SE polar region of the Homunculus, which views the minimum from a different direction. We used a model in which the He ii λ4686 emission is produced by two mechanisms: a) one linked to the intensity of the wind-wind collision which occurs along the whole orbit and is proportional to the inverse square of the separation between the companion stars; and b) the other produced by the ‘bore hole’ effect which occurs at phases across the periastron passage. The opacity (computed from 3D SPH simulations) as convolved with the emission reproduces the behavior of equivalent widths both for direct and reflected light. Our main results are: a) a demonstration that the He ii λ4686 light curve is exquisitely repeatable from cycle to cycle, contrary to previous claims for large changes; b) an accurate determination of the longitude of periastron, indicating that the secondary star is ‘behind’ the primary at periastron, a dispute extended over the past decade; c) a determination of the time of periastron passage, at ~4 days after the onset of the deep light curve minimum; and d) show that the minimum is simultaneous for observers at different lines of sight, indicating that it is not caused by an eclipse of the secondary star, but rather by the immersion of the wind-wind collision interior to the inner wind of the primary.