How apply the conformal mapping? (circle to rectangular)

Hello, I have a question about the conformal mapping in Matlab.

In the below case, I want to convert from the round segment closed by z1, z2, z3, z4 to the rectangular segment closed by z1' z2' z3' z4'.

How wirte the code? Thank you very much, and it would be a great pleasure for me.

Answer 1

[William Rose]

@Changwoo Lee,

Is this a homework problem?

The figure has an inconsistency. The left hand side shows a light blue region whose corners are z1, z2, and the origin. Points z3 and z4 are not corners of the light blue region on the left. On the right hand side, z3 and z4 are corners of the light blue region. Are you seeking a map that sends the light blue region on the left onto the light blue region on the right? Or are you seeking a map that sends (z1, z2, z3, z4) to (z1', z2', z3', z4')? I think it would be more clear if the light blue region on the left had the same corners as the light blue region on the right.

There are twio questions implied in your question:

1. What is the conformal map that does the desired mapping?
2. How can one implement the conformal map of quesiton 1 as a matlab script?

Do you know the answer to question 1?

[Torsten]

Do you know the answer to question 1?

Schwarz-Christoffel mapping:

https://arxiv.org/ftp/arxiv/papers/1509/1509.06344.pdf

Answer 2

[Torsten]

https://de.mathworks.com/matlabcentral/fileexchange/67117-rectangular-conformal-map

[William Rose]

@Changwoo Lee,

@Torsten has given you two excellent options for answering question 1. Actully the second option seems to answer both question 1 and question 2. . I recommend that you investigate the two options and decide which you want to pursue. Then follow up with specific quesitons here if and when you have them.

Good luck!

[William Rose]

@Changwoo Lee,

It is interesting that the standard polar to rectangular conversion fomula maps the curved segment to a rectangle, as shown below (script attached).

As you can see from the plot above, the right-angle intersections between radial lines and arcs in the curved segment get mapped to right angles in the rectangle. The preservation of some right angles in this transformation makes one hope that this mapping is conformal. Sadly, it is not. Most angles are not preserved under this mapping.

[Changwoo Lee]

@Torsten Thank you very much for the kind reply!

[Changwoo Lee]

@William Rose Thank you very much for the kind reply! I solved the problem. Your reply and code are a greate pleasure. Thank you again!

[William Rose]

@Changwoo Lee, You're welcome. Good luck with your work.