exupero's blog

Parabolas in taxicab geometry

In the previous post I showed how to calculate the distance between a point and a line in taxicab geometry. That's important for being able to draw a couple other shapes, in particular parabolas.

We're probably most familiar with parabolas as the shape made by a quadratic function, but as the Wikipedia article on parabolas notes, there are several different definitions of parabola that can all be shown to generate the same shape. The first definition in the article does rely on distance to define a parabola: it's the set of points equidistant from a point (the focus) and a line (the directrix). Here's an example of a parabola in taxicab geometry:

It has a U shape, similar to the parabolas we're used to. We can check that points on it are equal distances from the focus and directrix by drawing circles:

I only drew a circle at each vertex, but you can imagine what happens as you move the circle at the upper-left up its leg of the parabola: the distance to the directrix will grow vertically by the same amount as the distance to the focus, due to the definition of the Manhattan distance. The same goes for the other circle moving right on its leg of the parabola. For the segment between the two vertices, you can see what happens as a circle slides along it: one side touches the focus, the while the opposite side touches the directrix.

The example above is a special case of a parabola in taxicab geometry, one created when the directrix is 45° to the axes. When the directrix isn't at 45°, the ends of the parabola extend off in the same direction:

It still has a U shape but has gained a vertex. As the directrix angles from 45°, an added vertex slides in from infinity.

We can again use circles to check the vertices' distances to the focus and directrix:

To construct a parabola under taxicab geometry, the easiest way I know is to find two specific points on the directrix, one that extends from the focus along a 45° angle upward, and one from the focus on a 45° downward. You can see those points at the bottom corners of two of the outermost circles above, but here they are more clearly, labeled A and B:


The outer vertices of the parabola are aligned in one direction with A or B, and in the other direction with the focus:


The center vertex of the parabola is halfway between the focus and the closest point of the directrix:


In the case of a directrix at 45°, the two vertices are halfway between the focus and directrix in both the horizontal and vertical directions:

This post has mostly been about the construction of parabolas under taxicab geometry, since I'm not aware of interesting properties of such parabolas. If you know of any, please email me.