Directrix-based hyperbolas in taxicab geometry

In the previous post I illustrated directrix-based ellipses in taxicab geometry, which are conic sections, while sum-based ellipses are not. The same lesson applies to my original post on taxicab hyperbolas. There, I drew hyperbolas as the points that are a fixed difference of distances from two foci, but there's also a directrix-based definition of hyperbolas. In Euclidean geometry, the two definitions produce identical shapes, but different shapes in taxicab geometry.

The definition of all conic sections is the same: the ratio of the distance between a point and a focus to the distance between the point and a line. That ratio is called the eccentricity, and what distinguishes conic sections from each other is not their definition but the ranges of their eccentricities. A circle has an eccentricity of 0, ellipses an eccentricity between 0 and 1, parabolas an eccentricity of 1, and hyperbolas an eccentricity greater than 1.

Hyperbolas still have two branches, but they're not symmetrical.

Here are the same three types of hyperbola, shown as sections of a double pyramid:

The ranges of eccentricies for the various conic sections correspond to the slope of the cutting plane in relation to the cone's sides. A circle, with eccentricity 0, is produced by a flat cutting plane. An ellipse, with eccentricity between 0 and 1, is produced by a non-flat cutting plane that is less sloped than the sides of the cone. A parabola, with eccentricity 1, is produced by a cutting plane with the same slope as the sides of the cone. And a hyperbola, with eccentricity greater than 1, is produced by a cutting plane that's steeper than the sides of the cone, causing it to intersect both halves of a double cone and thus have two branches.

In our more Euclidean reality, conic sections appear in gravitational mechanics, with elliptical orbits and hyperbolic flybys, which raises the question of how a spacetime using the taxicab metric would behave. If you have ideas, please let me know.

For more on conic sections in taxicab geometry, there's a 1982 paper by Richard Laatsch. I haven't found a public version, but if you know of one, feel free to email me.