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Variations on a Theme of  Real and Imaginary Linear Measure

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.
The 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 intervals on that line
    freed from restraint (drag the yellow point)
    as the two invariants go imaginary,

  • then,

  • 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 conics.

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 two perspectivities in the directrices on the green line participating in a projectivity
that always forms intervals whatever the state of these invariants - real, or imaginary.