exupero's blog

Ellipses in taxicab geometry

In the previous post I used the Manhattan distance (instead of Euclidean distance) to make a circle that was also a square. A circle, however, is a degenerate form of an ellipse. An ellipse has two foci, and a circle appears when both foci are in the same place.

In taxicab geometry, separating the two foci horizontally or vertically turns a four-sided circle into a hexagon:


But hexagonal ellipses are also a degenerate form. The fullest form of taxicab geometry's two-dimensional ellipse is an octagon, which occurs when the foci are neither vertically nor horizontally aligned:


Of course, a two-dimensional ellipse is a degenerate form of a three-dimensional elliptical solid, having both foci in the same axial plane. While a sphere in taxicab geometry is an octahedron, an elliptical solid is a more complex shape which I find easy to visualize but have yet to find the proper name for. If you know it, or you know the Conway polyhedra notation for it, please email me.