exupero's blog

Hyperbolas in taxicab geometry

In the previous post I showed how parabolas are defined using distance and how they look in taxicab geometry. Another algebraic shape that can be defined using distance is the hyperbola.

A hyperbola is the set of points where each point has an equal difference of distances to two foci. In taxicab geometry, one type of hyperbola looks like this:

Each point on the two lines above is 6 units further from one focus than the other focus. We could have chosen some other distance, but when the foci are aligned horizontally or vertically, and the chosen distance is less than the distance between the foci, then the hyperbola will always be two straight, parallel lines.

When the difference of distances equals than the distance between the foci, the shape does something we haven't seen so far in taxicab geometry: it becomes an area:

All the points in the shaded areas are 10 units further from one focus than the other. No difference of distances greater than 10 units is possible.

Parallel lines aren't the only way hyperbolas look under taxicab geometry. If the foci aren't aligned horizontally or vertically, they can bend:

All the points on the above lines are 4 more units from one focus than the other. To develop a bit of intuition about why, consider a pair of circles centered on the foci, with radii of 5 units and 9 units (a difference of 4 units):

The diagonal portions of the hyperbola lie where the two circles meet. Where the circles separate at a corner, the hyperbola bisects the right angle and falls on the vertical. The hyperbolas extend vertically because the foci are farther apart horizontally than vertically. If the vertical distance between the foci was bigger than the horizontal distance, the hyperbola would extend horizontally. You might be able to see why if you imagine sliding the centers of the circles around, watching where they meet and separate.

If you do mentally slide the blue circles around, you may realize that I had to choose the above example somewhat carefully. Hyperbolas in taxicab geometry have a lot of edge cases. When the difference of distances is small enough, we get two parabolas:

The logic about where the hyperbolas appear hasn't changed, only the geometric relationship between the reference circles.

So what happens when a corner of one circle meets the corner of the other circle?

Each component of the hyperbola has a portion that flares out into an area, in which each point has the same difference of distances to the foci, in the above case, 6 units.

Another edge case occurs when, again, the difference of distances is equal to the distance between the foci:

A couple other edge cases are more properly termed degenerate cases, which occur when the difference of distances is zero and a hyperbola's separate components merge into one:

If the foci are the same distance apart horizontally as vertically, we get a barbell-shaped hyperbola:

These last two forms we'll see again later in this series.

Those are all the ways hyperbolas can look under taxicab geometry, at least that I've found so far. If you can think of more, let me know.