exupero's blog

Conic sections in taxicab geometry

So far all the shapes we've considered under taxicab geometry are, in Euclidean geometry, conic sections: circles, ellipses, parabolas, and hyperbolas. Let's see if these shapes are conic sections in taxicab geometry as well.

Not all cones are based on a circle, but the one used for Euclidean conic sections is. Since a circle in taxicab geometry is a square, a taxicab cone is a pyramid:

Plainly taxicab circles are conic sections:

Two-vertex parabolas might be a conic section:

However, not just any such section will create a parabola. Only certain angles of the cutting plane or certain angles of the cone produce 45° angles at the vertices. Maybe the cutting plane has to be parallel to one side of the cone/pyramid? Perhaps the pyramid has to have a 45° or 90° apex?

Three-vertex parabolas also seem possible, with the requirement that the end rays extend parallel to each other:

It's hard to imagine how a hyperbola could be a conic section. Hyperbolas in taxicab geometry take eight different forms, but half of them have some portion that fans out into an area. The only way a cutting plane produces an area on a pyramid is when it aligns with one of the pyramid's faces:

For that to be a hyperbola, the sides of the cone must meet at the apex at 90°.

In addition to area problem, I don't see anywhere for a line to extend infinitely in two directions, let alone with two 45° bends in the middle. The closest we can come to a hyperbola is as a variation of the parabola above:

That uses a cutting plane parallel to the cone's axis of symmetry, as done when creating a Euclidean hyperbola. In Euclidean geometry the two components of a hyperbola are created by using a double cone, with the cutting plane intersecting both of them. We can try that in taxicab geometry too:

It's hard to see, but the alignment doesn't quite match the form of hyperbola that it looks like it does. The cones need to be offset, something like this:

Possibly offsetting the cones could address some of the peculiarities of hyperbolic conic sections, but it feels like adding epicycles to a bad model.

The only other conic section is the ellipse. I don't see any way to get an hexagon or octagon out of a pyramidal conic section. Euclidean ellipses are made by tilting the cutting plane from horizontal, but doing so in taxicab geometry doesn't add sides, it changes their lengths:

Geometrically, that's a kite.

It feels safe to say that taxicab shapes aren't conic sections, but if you have further information about conic sections under taxicab geometry, please email me.