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

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.

• Not a "metric" variant, but: Switch to the "unit hyperbola" ($x^2 - y^2 = 1$) and define the hyperbolic trig functions. Further, with $P$ on the curve, $Q$ at $(1,0)$ and $O$ the origin, define the "hyperbolic radian measure" of the angle $POQ$ by "twice the area of sector $POQ$", which matches the circular case. Indeed, when you get to integrating $1/x$ (which is an hyperbola of "radius" $\sqrt{2}$ rotated by 45 degrees), you can (w/work) express that area in terms of logs and, in turn, derive the standard formulas for the hyp-trig functions in terms of the exponential of hyperbolic radians. – Blue Nov 4 '10 at 11:31
• how about the lorentz metric $\mathbb{R}^{3,1}$. – Matt Calhoun Nov 13 '10 at 14:00
• I really like this question, both the goal and the subject. +1 – BBischof Nov 13 '10 at 21:02

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"?

• @Agustí Roig: These two are definitely interesting, but as applied to what a little don wants, they just give you "old" answers: the distance to the origin in the elevator distance is the taxi-cab norm, and the distance to the origin in the mail office distance is the usual distance. For the former, the "unit circle" is the rectangle $(1,0)(0,1)(-1,0)(0,-1)$, and for the latter the usual unit circle... – Arturo Magidin Nov 4 '10 at 4:18
• @Arturo. You're right. I'm correcting this. And, you're right too, the distance to the origin is the taxicab norm, but not for other points in $\mathbb{R}^2$: it's the way to measure the distance when you go from the 53th floor of one Petronas Tower to the 84th floor on the other one. :-) – d.t. Nov 4 '10 at 4:23
• @Agustí Roig: Yes, I did figure out the answer to your last question, which is how I noticed you had your components switched. (-: – Arturo Magidin Nov 4 '10 at 4:26
• @Arturo. But, again, the interesting part are the other circles, centered at other points, not the origin. I particularly like the mail office distance as an example of how a metric space doesn't need to be second countable. I'll add your remark about the balls centered at the origin. – d.t. Nov 4 '10 at 4:27
• With these metrics, the shape of a circle depends both on its centre and on its radius. This could be another interesting aspect for don's students to explore. – Rahul Nov 4 '10 at 5:13

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.

• We have talked about p-norms, I think the sup norm might be a bit confusing since they just learned about infinity this semester. But, it's worth a shot. I need to see what linear transformations do the unit circle...I don't know off of the top of my head. (I'm hoping for a "cool" unit circle) – futurebird Nov 4 '10 at 3:17
• @a little don: The reason you have an $\infty$ symbol in the sup norm is just that the sup norm is the limit, as $p$ goes to infinity, of the $p$-norms. There is absolutely no need to evne mention infinity in defining the sup norm: it's just the maximum of the absolute values of the entries, no $\infty$ at play at all. As for linear transformations, if they have two eigenvalues, you end up "distorting" the unit circle along the corresponding eigenspaces. Take them orthogonal and you get ellipses. – Arturo Magidin Nov 4 '10 at 3:20
• ellipses sound very "cool" thank you, Arturo. – futurebird Nov 4 '10 at 3:39
• @don: On the other hand, developing the equivalents of "sine" and "cosine" would seem to be a tad intricate with the Hölder norms. As for seeing what the "unit circles" might look like... – J. M. is a poor mathematician Nov 4 '10 at 3:42
• Nitpick: the p-norms are only defined for p >= 1. For 0 < p < 1, the unit balls are not convex and thus, do not define a norm. – G. Rodrigues Nov 5 '10 at 12:28

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.

• I would like to follow this up. Where can I find out more? – MJD Dec 4 '13 at 15:35

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 $2$.

• You write p, q colinear; is that p, q, 0 collinear, i.e. the line pq passes through the origin? – Charles Nov 4 '10 at 13:21
• @Charles: yes (it would be a bit of a daft definition otherwise!). – Loop Space Nov 4 '10 at 13:31