Point-line difference shapes

Another shape without a common name is the one defined as a constant difference of distances to a focus and a directrix:

In Euclidean geometry, these appear to be a parabola:

That makes sense, since a parabola is the set of points equidistant from a focus and a directrix, or having zero difference in distances.

Interestingly, though, this definition can also produce a kind of double parabola:

Taxicab geometry has analogs for both:

The double parabola above just looks like two angled lines, and to see them as overlapping parabloas requires some foreknowledge of what you're looking for.

With the directrix at 45° to the axes, the shape still looks like a taxicab parabola with a similarly sloped directrix:

When forming a double parabola against a directrix at 45°, the shape becomes two right angles:

With a sloped line, some foreknowledge again helps to see the underlying pair of overlapping parabolas: