You can drag all the intervals on the bold black line by using the yellow point
on the bold conic, and can drag the green and black lines independently.
Invariants are the
intersections of the conic and the black line; the Directrices are the
intersections of the conic and the green line.
First, drag the
bold black line clear of the conic,
to see the
on that line
freed from restraint (drag the yellow point)
as the two invariants go imaginary,
drag the bold green line clear of the conic,
to see those intervals, and the net that formed them, go imaginary.
Do note that the
now-imaginary net still forms real
The key insights here are that equivalent or identical intervals are formed
on all the conics,
and that the defining black and green lines form the degenerate conic of the set.
Thus we have
in the directrices on the green line participating in a projectivity
always forms intervals
whatever the state of these invariants - real, or imaginary.