Timeline for Reference request: affine transforms + circle inversion?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 19, 2013 at 17:02 | comment | added | Bill Bradley | Thank you, that's very helpful! Thinking about that zoo of curves makes it even more remarkable that circles are preserved by Mobius transformations. | |
| May 18, 2013 at 5:53 | comment | added | Douglas Zare | Circle inversion doesn't preserve generalized ellipses, or conics. Curves of degree n are typically sent to curves of degree 2n. If you invert an hyperbolas about its center you get a figure-8s, a lemniscates. Ellipses can be sent to dimpled limaçons or hippopedes. Parabolas can be sent to cissoids or cardiods. If the center of inversion is on the conic, though you get a cubic curve like a crunode $y^2 = x^2(x+1)$. en.wikipedia.org/wiki/Inverse_curve xahlee.info/SpecialPlaneCurves_dir/Inversion_dir/inversion.html | |
| May 17, 2013 at 22:23 | comment | added | Misha | Bill: You cannot, since you have to take compositions as well. | |
| May 17, 2013 at 21:50 | answer | added | Robert Bryant | timeline score: 6 | |
| May 17, 2013 at 21:03 | comment | added | Bill Bradley | I think so-- I believe you can write any such transformation as either $b+A(x-c)/|A(x-c)|$ or $b+A(x-c)$, for $A\in GL_n(R)$. | |
| May 17, 2013 at 17:45 | comment | added | Anton Petrunin | Do you know if $\cal{T}$ is finite-imensional? | |
| May 17, 2013 at 13:47 | history | asked | Bill Bradley | CC BY-SA 3.0 |