Change the coordinate system by embedding it on a sphere; plot the hyperbole on the sphere where now the point at infinity of the hyperbole passes through zero on the other side of the sphere (where x recrosses y). I take this as a diagram of the discontinuity at zero in passing from -1 to 1 of the equation. It is the backside place of the coordinates usually not draw in high school texts that perhaps can be called the 'imaginarea'. If this is the case, then the discontinuity at the backside 0 point on the sphere can be removed by remapping the hyperbole and cartesian coordinates onto the projective plane ... no? In which case, we discover the discontinuity involved is a twist of the infamous Mobius band.

Thanks for your article,

Scully