Timeline for Conformal map onto a circle, from an identification space composed of five squares
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 21, 2019 at 14:36 | comment | added | Alexandre Eremenko | It says that the side of the trapezoid between the vertices with angles $\pi/4$ and $\pi/2$ is three times longer than the side between the vertices with angles $\pi/2,\pi/2$. Together with given angles, this condition completely describes the shape of the trapezoid, up to similarity. And multiple $i$ in included because these sides are perpendicular. | |
| Dec 21, 2019 at 14:09 | comment | added | niran90 | Okay thanks, could you also explain where the second equation for the accessory parameter stems from? | |
| Dec 19, 2019 at 3:17 | comment | added | Alexandre Eremenko | It is the preimage of one of the vertices. | |
| Nov 27, 2019 at 1:04 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
added 1 character in body
|
| Nov 26, 2019 at 16:55 | comment | added | niran90 | could you explain the geometric significance of the accessory parameter in the context of the mapping, as well as that of the second equation in your answer? | |
| Nov 25, 2019 at 13:49 | vote | accept | niran90 | ||
| Nov 24, 2019 at 15:20 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
added 244 characters in body
|
| Nov 24, 2019 at 15:12 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
deleted 508 characters in body
|
| Nov 24, 2019 at 13:53 | comment | added | Alexandre Eremenko | Schwarz-Christoffel formula is derived or explained in most complex analysis textbooks. Of course you need some background on complex analysis which can be found in the same books. Take any of those books which has "Schwarz Symmetry Principle" and "Schwarz-Christoffel formula". | |
| Nov 24, 2019 at 5:39 | comment | added | niran90 | Okay cool, got it. Apologies - that's my lack of mathematical background talking. Would the reference you've provided be a good start for learning about how to derive the mapping function? Or would I actually need a firm background in complex analysis first? I have to be honest in that, most of the concepts you mention in the latter part go straight over my head. | |
| Nov 24, 2019 at 4:01 | comment | added | Alexandre Eremenko | "Conformal" means of course "conformal inside the region, not on the boundary. Otherwise how could you map even one square on a disk?? | |
| Nov 23, 2019 at 23:51 | comment | added | niran90 | I think just grasped the former part of your post. Do you think by vitue of the fact that 3PI/4 angle is mapped onto PI, the map cannot be conformal? Also, I am not entirely familiar with Christoffel-Schwartz yet, but what would be the procedure to derive the function that takes in a coordinate from the original domain and spits out a coordinate in the mapped space? | |
| Nov 23, 2019 at 22:44 | comment | added | niran90 | Thanks so much for this info! I am not sure if I'm completely clear on some of the ideas though - do you mean 8 trapezoids or 8 right-angled triangles with angles 45,45,90? And if so, does the procedure you've proposed above ensure angle-preservation at the boundaries of the sectors too? I require that all angles, except for the ones surrounding the 4 points at which three square regions meet, preserve their orthogonality. | |
| Nov 22, 2019 at 18:13 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
deleted 1682 characters in body
|
| Nov 22, 2019 at 17:58 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
added 987 characters in body
|
| Nov 22, 2019 at 15:37 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
added 300 characters in body
|
| Nov 22, 2019 at 14:22 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
added 104 characters in body
|
| Nov 22, 2019 at 14:12 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
added 230 characters in body
|
| Nov 22, 2019 at 13:57 | history | answered | Alexandre Eremenko | CC BY-SA 4.0 |