The polar line is the locus of the intersections of tangent lines at the ends of chords of teh parabola through a fixed point.
Turns out to be conceptually important - not just a curiosity.

I saw a demo where a foam cube had a knitting needle inserted along the long diagonal, and was rotated by spinning the needle between the palms of the hands.
The sides of the cube formed a curve... in fact a portion of a hyperbola, as this video illustrates.