Geometry Taxicab geometry

Point-line product shapes

The last of these unnamed shape definitions is the one that's a constant product of distances to a focus and a directrix:

These shapes are cousins of the ovals of Cassini, which are a constant product of distances to two foci. Here, though, one of the foci is a line, so half the shape wraps a focus that extends to infinity:

A constant product of distances to a point and a line, in Euclidean geometry.

Similarly for taxicab geometry:

A constant product of distances to a point and a line, in taxicab geometry.

Both can be fully disconnected:

Two disconnected components, in Euclidean geometry.

Two disconnected components, in taxicab geometry.

or fully joined:

Two connected components, in Euclidean geometry.

Two connected components, in taxicab geometry.

Interesting things happen to the taxicab shape when the directrix is neither horizontal or vertical. Here it is at 45° to the axes:

With the directrix at 45° to the axes, components just touching.

With a sloped directrix, the shape becomes asymmetric (and, to my imagination, birdlike):

September 2025