Timeline for answer to What is rotation when we have a different distance metric? by Zach466920
Current License: CC BY-SA 3.0
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| May 4, 2022 at 6:12 | comment | added | paperskilltrees | Can this definition of angles be generalised to higher dimensions? It looks like the angle between any two vectors on a coordinate plane is well-defined, but what about two arbitrary vectors that are non-coplanar with coordinate axes in higher dimensions? The Eucledian angle is tied to the scalar product; 2-norm is the only $p$-norm associated with an inner product; hence the angle you defined has to have fewer properties than the "true" angle. I wonder where exactly it breaks? Any thoughts? | |
| Dec 30, 2016 at 12:29 | vote | accept | Ian Miller | ||
| Dec 16, 2016 at 6:56 | vote | accept | Ian Miller | ||
| Dec 30, 2016 at 12:29 | |||||
| Dec 14, 2016 at 16:28 | comment | added | symplectomorphic | @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. | |
| Dec 14, 2016 at 2:33 | comment | added | Ian Miller | 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. | |
| Dec 14, 2016 at 0:39 | comment | added | Ian Miller | 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? | |
| Dec 14, 2016 at 0:31 | comment | added | Zach466920 | 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. | |
| Dec 14, 2016 at 0:27 | comment | added | Ian Miller | 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. | |
| Dec 13, 2016 at 21:29 | history | edited | Zach466920 | CC BY-SA 3.0 |
added 1 character in body
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| Dec 13, 2016 at 21:16 | history | answered | Zach466920 | CC BY-SA 3.0 |