Iterative Tangents RSS feed

A curiosity journal of math, physics, programming, astronomy, and more.

Geometry Taxicab geometry

Point-line sum shapes

In previous posts about taxicab geometry, I've illustrated shapes which have common names: ellipse, parabola, hyperbola, oval of Cassini, and circle of Apollonius. Occasionally I've had to distinguish between shapes which have different definitions but the same name in Euclidean geometry, because they present distinctly in taxicab geometry, specifically sum-based ellipses versus directrix-based ellipses, and difference-based hyperbolas versus directrix-based hyperbolas. Now, however, we turn to shapes that, as far as I can tell, don't have common names. (If you know a name for these shapes, please email me.)

The first is a shape defined as a constant sum of the distances to a focus and a directrix:

d i s t left-parenthesis p comma f right-parenthesis plus d i s t left-parenthesis p comma l right-parenthesis equals c

In Euclidean geometry, these are lens-shaped but asymmetric:

A shape defined by a constant sum of distances to a point and a line, in Euclidean geometry.
A shape defined by a constant sum of distances to a point and a line, in Euclidean geometry.

Taxicab geometry's versions seem to be hexagonal approximations of the Euclidean shape:

A shape defined by a constant sum of distances to a point and a line, in taxicab geometry.
A shape defined by a constant sum of distances to a point and a line, in taxicab geometry.
With a directrix at 45° to the axes.
With a directrix at 45° to the axes.
With a directrix neither vertical, horizontal, nor at 45° to the axes.
With a directrix neither vertical, horizontal, nor at 45° to the axes.

The only other case seems to be the occasional pentagon:

A five-sided shape defined by a constant sum of distances to a point and a line, in taxicab geometry.
A five-sided shape defined by a constant sum of distances to a point and a line, in taxicab geometry.