Timeline for Functions $\mathbb{R}^2\to\mathbb{R}^2$ that preserve lines
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 29, 2022 at 17:49 | vote | accept | Robin Houston | ||
| Apr 25, 2022 at 9:15 | answer | added | Adam P. Goucher | timeline score: 3 | |
| Apr 24, 2022 at 20:00 | history | edited | Robin Houston | CC BY-SA 4.0 |
Clarify that Putman’s note proves the general case, not just the specific case I’m using in this question
|
| Apr 24, 2022 at 19:25 | history | edited | Robin Houston | CC BY-SA 4.0 |
Explain some consequences of the assumption that the image of $f$ contains three non-collinear points
|
| Apr 24, 2022 at 18:04 | comment | added | Robin Houston | I did originally go through your proof reasonably carefully, specialising it to this case, and I didn’t seem to run into any obstacles to doing so | |
| Apr 24, 2022 at 17:46 | comment | added | Andy Putman | @RobinHouston: That probably works, though I would have to think harder to make sure there are no subtle issues. | |
| Apr 24, 2022 at 16:45 | comment | added | Robin Houston | @AndyPutman I think my claim follows fairly easily from the n=3 case of your theorem, identifying $\mathbb{R}^2$ with the plane $z=1$ in $\mathbb{R}^3$ etc. But if you’re saying it doesn’t, I’ll check that more carefully! | |
| Apr 24, 2022 at 16:14 | comment | added | Andy Putman | My note doesn’t prove what you assert: it is about automorphisms of the set of linear subspaces that preserve incidence relations, and what it proves is false for n=2. | |
| Apr 24, 2022 at 16:09 | history | edited | Robin Houston | CC BY-SA 4.0 |
Acknowledge @Wojowu’s comment
|
| Apr 24, 2022 at 16:04 | comment | added | Robin Houston | Oh, thanks @Wojowu! I wondered if there was an example of that general sort, but I wasn’t sneaky enough to think of multiplying by the sin. I’ll edit the question. | |
| Apr 24, 2022 at 15:54 | comment | added | Wojowu | The answer to your last question is no. There exists a function $f:\mathbb R^2\to\mathbb R$ with the property that its restriction to any line is surjective (for instance, $f(p)=|p|\cdot\sin|p|$), and then $g:p\mapsto(f(p),0)$ is very non-affine and maps every line to the $x$ axis. | |
| Apr 24, 2022 at 15:49 | comment | added | LSpice | An old question of mine that's related: Non-affine, projective vector field on $\mathbb R^n$. See particularly @SergeiIvanov's answer (though it may just be re-proving what you already know). | |
| Apr 24, 2022 at 15:44 | history | asked | Robin Houston | CC BY-SA 4.0 |