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

Question

My class had a good time redeveloping the unit circle under the taxicab metric. Now some of them want to do it again with another similar metric. I want to give this to some of my "honors" high-school* calculus I students to work on independently. As such, it can't be too difficult or require too many advanced concepts.

What is another metric that I could introduce them to where one can describe a "unit circle" and try to make sense of the sine, cosine and tangent functions?

*I have a group of high school students enrolled in a college course. The course, as planned, is far too easy for them. (At the same time, it is quite hard for the college students enrolled. Go figure.) After the high school students do what has been assigned by the online learning system I have them work on other stuff-- which I allow to be pretty self-guided.

Answer 1

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"?

Answer 2

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.

Answer 3

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 $2$.

Answer 4

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.