Developing the unit circle in geometries with different metrics: beyond taxi cabs

Two funny distances on $\mathbb{R}^2$:

The elevator distance.

$$ d((x_1,y_1),(x_2,y_2)) = \begin{cases} \vert y_1 - y_2 \vert & \text{if}\ x_1 = x_2 \\ \vert y_1 \vert + \vert x_1 - x_2 \vert + \vert y_2\vert & \text{otherwise} \end{cases} $$

The mail office distance.

$$ d(p,q) = \begin{cases} 0 & \text{if}\ p = q \\ \|p\| + \|q\| & \text{otherwise} \end{cases} $$

A good question to ask: why are they called "elevator" and "mail office" distances, respectively? :-)

EDIT. As Arturo Magidin points out, with these distances balls centered at the origin are not particularly interesting: you have to try with balls NOT centered at the origin.

MORE EDIT. Don: have you seen Arturo Magidin's comment about making balls centered at the origin "interesting"?

Any norm defines a distance, by $d((a,b),(c,d)) = ||(a-c,b-d)||$. Some common norms:

1. There are all the $p$-norms: $||(x,y)||_p = \sqrt[p]{|x|^p + |y|^p}$ (the usual norm occurs with $p=2$); you can do them for any $p$, $0\lt p\lt \infty$.

2. The sup norm: $||(x,y)||_{\infty} = \max\{|x|,|y|\}$.

3. You can take a positive linear combination of norms to create a new one.

4. Given any linear transformation $A\colon\mathbb{R}^2\to\mathbb{R}^2$, and any norm $||\cdot||$, you can define the norm that maps $(x,y)$ to $||A(x,y)||$.

There is also the discrete metric, though that will create a rather nasty "unit circle".

It is actually nice to treat the sup norm, the $p$-norms, and the taxi-cab norm together. If you draw the "unit circles" for all of them, the limit of the $p$ norms as $p\to 0^+$ is the taxicab norm, while the limit as $p\to\infty$ is the sup norm.

It is a lovely result of Hermann Minkowski that any plane centrally symmetric convex set can serve as the "unit ball" of a distance function.

One closely related to the two in Agusti's answer is what I call the bus metric (though I change the name to reflect the name of the local bus company - which seems to change each time I teach about metric spaces!). This is:

$$ d(p,q) = \begin{cases} \|p - q\|_2 & \text{if } p, q, 0 \text{ are collinear} \\\\ \|p\|_2 + \|q\|_2 & \text{otherwise} \end{cases} $$

(In Trondheim (where I am) then the majority of bus routes are radial, so this does link to the students' intuition.)

The students could do a nice animation of what happens to a ball of unit length centred at a point $(x,0)$ as $x$ ranges from, say, $-2$ to $2$.