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Background

I am looking at the Taxicab metric in 2D and other related norms of the form:

$$d(\vec{x}, \vec{y}) = \|\vec{x} - \vec{y}\|_p = \left( \sum_{i = 0}^n |x_i - y_i|^p \right)^{1/p}$$

I haven't formally studied this topic so am investigating ideas from the Euclidean norm and applying them to other values of $p$. In a previous question I looked at how the circumference of a circle (a set of points equal distance from a given point) changed with $p$. The best answer has now made me question what do we mean by 'rotation' as it talked about rotational-invariance but wikipedia's page discussing the topic refers to trigonometric functions which have strong links to the Euclidean measure.

In Euclidean geometry my understanding of an angle is the ratio of the length of a arc to the radius of that arc. Minor Question: Is this definition suitable to apply to other norms?

I applied this definition to the Taxicab metric and definited $\cos_1(\theta)$ and $\sin_1(\theta)$ to be the $x$ and $y$ coodinates of the point on a unit circle with angle $\theta$ (note in Taxicab a circle looks like a diamond). My graphs look like this (note that $\pi_1=4$ for Taxicab geometry):

enter image description here

Next I decided to investigate a rotating object with the Taxicab metric to try to understand the idea of rotational invariance. I took a number of points defining the perimeter of a unit square and rotated it. This is what I got:

enter image description here

At first I thought this was very strange until I draw in concentric circles going through every point I used to represent my square.

enter image description here

This demonstrated that each point is simply following a 'circular' path which is what we expect for a rotation. So perhaps my brain is just so used to looking at Euclidean rotation it can't accept that. (As an anology watching a tesseract rotate looks like it is distorting and bending but it just my inability to view it as a 4D object). So is this square rotating correctly? Or is this an example of rotational invarience?

Question

  • How it rotation defined for other metrics/norms/vector spaces?

  • How is rotational invarience defined/calculated in other metrics/norms/vector spaces?

Note

I'm still coming to grips with the notation/language so may have misused words/symbols such as norm/metric/etc.

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  • $\begingroup$ @symplectomorphic It looks even trippier when I apply a normal Euclidean rotation in the opposite direction. The square just ungulates. I'd love if you could write an answer expanding upon what you mean by not including every rotation. Using my definition of angle above I feel I could rotation by any amount. $\endgroup$ – Ian Miller Dec 14 '16 at 0:51
  • $\begingroup$ I deleted my first comment because I thought it was mistaken, but then I realized it isn't. The symmetry group (= group of isometries) of the unit circle in the taxicab plane contains only four rotations. (So it is indeed discrete, unlike $SO(2)$.) It is true that rotating the unit circle by any angle maps the unit circle to itself, but not necessarily isometrically; our definition of a rotation better make it an isometry. Only four rotations map the circle to itself isometrically: rotations by "angles" 2, 4, 6, and 8. See the nice paper dergiler.ankara.edu.tr/dergiler/29/1396/15831.pdf $\endgroup$ – symplectomorphic Dec 14 '16 at 16:23
  • $\begingroup$ This paper also looks interesting, but I haven't looked yet: jstor.org/stable/2322995?seq=1#page_scan_tab_contents $\endgroup$ – symplectomorphic Dec 14 '16 at 16:48
  • $\begingroup$ One last interesting reference: page 48 here, which gives the nice result that the group of linear isometries of the plane with determinant $\pm1$ is finite iff the norm is not induced by an inner product. (The taxicab norm is one such norm.) $\endgroup$ – symplectomorphic Dec 14 '16 at 21:07
  • $\begingroup$ I lied; I have one more reference. Here is a deep way of asking and answering your question, from MO. Sorry for all the comments, but these are nice questions I never asked explicitly: I've found them very fun to think about. $\endgroup$ – symplectomorphic Dec 14 '16 at 21:16
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How it rotation defined for other metrics/norms/vector spaces?

Some definitions:

$(1)$ Let $C_d$ be the unit circle for a metric $d((x_1,y_1);(x_2,y_2))$.

$(2)$ Let the arc length of $C_d$, with respect to $d$, equal $2 \cdot \pi_d$.

$(3)$ Given two rays that extend from the origin to $C_d$ the angle $\theta_d$ between them is the arc length of the circle contained in the two rays.

Angle

$(4)$ We can define $\cos_d(\theta_d)$ as the $x$ value if an angle $\theta_d$ is swept starting from the right and going counter clockwise. Similarly, $sin_d(\theta_d)$ as the $y$ value if an angle $\theta_d$ is swept. In other words the two functions allow us to rotate points about $C_d$.

Results:

Clearly we have $$d((\cos_d(\theta_d),0);(0,\sin_d(\theta_d))=1$$

If we wish to rotate a point $r_1=(x_1,y_1)$ we need to use a matrix $R_d$. Then we can obtain the location of the rotated point as $r_2=(x_2, y_2)$ using,

$$\vec r_1=R_d \cdot \vec r_2$$

$$R_d=\begin{bmatrix} \cos_d(\theta_d) & -\sin_d(\theta_d) \\ \sin_d(\theta_d) & \cos_d(\theta_d) \end{bmatrix}$$

Note that rotation doesn't change the length with respect to the metric $d$. Now we're ready to talk about invariance.

How is rotational invarience defined/calculated in other metrics/norms/vector spaces?

The first thing is that rotation of the unit circle results in another unit circle. In fact, we have,

$$C_d=R_d \cdot C_d$$

Below we have the unit circle for the $p=1$ p-norm metric. We see that at various stages of rotation we still have the same unit circle since it's invariant under rotation. However, if we rotate one part of the circle, it doesn't remain invariant.

Rotation

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  • $\begingroup$ Why does the bottom pictures mean it is invariant? It appears we are applying our subjective view of what variation means. The red part looks different in Euclidean space but every point on the red arc has exactly the same radius (just changed angle) so a person used to Taxicab space would see it as not changing. $\endgroup$ – Ian Miller Dec 14 '16 at 0:27
  • $\begingroup$ The bottom picture shows that the entire circle is invariant to rotation. However, it also shows that arcs along the circle are not invariant. You can do the same thing with a Euclidean circle. Applying a rotation leaves the object unchanged, so it's invariant. However, arcs along the circle are rotated to other parts along the circle. $\endgroup$ – Zach466920 Dec 14 '16 at 0:31
  • $\begingroup$ Ok, thanks, I misunderstood. The picture is showing that an arc varies but a circle doesn't. Can you expand upon this statement you had in your previous answer "In other words, it is rotation invariant. Try it, you can rotate the coordinate system without changing the length. You can't do that with the Taxicab metric, the length you get will change." How can a rotation with the Taxicab metric cause the length to change? $\endgroup$ – Ian Miller Dec 14 '16 at 0:39
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    $\begingroup$ I think I've partially worked out something. In your three pictures the length between the end points changes. In the first and last picture the length between end points is 2 while the middle one has a a distance between end points of only 1.This doesn't happen in Euclidean rotation. $\endgroup$ – Ian Miller Dec 14 '16 at 2:33
  • $\begingroup$ @Ian Miller: the point you make in your last comment here explains why the symmetry group of the taxicab unit circle contains only four rotations, not all of them. Most taxicab "rotations" (maps that move points along taxicab circles) are not isometries, even though they all map the unit circle onto itself. This takes some retraining of your intuition. $\endgroup$ – symplectomorphic Dec 14 '16 at 16:28

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