If you are willing to discretize some of the more complicated geometries then I suppose chaining Schwarz–Christoffel maps between the disk and the upper half plane with en.wikipedia.org/wiki/Schwarz%E2%80%93Christoffel_mapping would work although you would need the inverse map which might lead to branch cut issues and might not be conformal in certain regions.

This is rather math, not Mathematica.

Look in mathoverflow.net/questions/314189/… for info.

@user64494 It is no doubt a math problem, but I want to use MMA to solve this math problem.

@userrandrand I'm glad you tried, but both of my regions are simply connected. According to theory, isn't it certain that a biholomorphic map can be found?

@user64494 Yes, I read it, in fact I retrieved your post before I asked the question. But I was trying to find this conformal mapping using MMA, that's why I came here to ask the question

@yode I do not know what the general algorithm is for an arbitrary geometry but user64494's link looks pretty good although I did not check the code. I should also mention that I believe that the integrals involved in calculating the Schwarz–Christoffel maps are rather slow to compute but this is from memory reading about a method to optimize integrals. In that method, they decided to look for approximations of the Schwarz–Christoffel maps if I remember correctly, the methods in the link by user64494 also seem approximate but I did not read the posts.

In your case, it seems that you would like to find the map given explicit equations for the regions. I do not know what can be done in general outside of polygons but you could maybe ask at math stack exchange.

@yode: I repeat my question "What did you try on your own?". Waiting for a serious reply of you.

@user64494 I don't know how to do, so I ask here...

@yode: Can you first set up Schawarz-Christoffe integral via NIntegrate between upper half-plane and the red region discretized into a polygon? Just use a few vertices initially to see how NIntegrate handles it.

@josh Does Schawarz-Christoffe only seems to apply to the connected region of the polygon?

@yode: I looked into this problem when you posted it. However, Schawarz-Christoffe in general is difficult to code; my prev. experience with it was a specialized simple case. I thought the reference given by user64494 would be useful for a more thorough investigation. Start small with a simple case and build up from there.

@yode: Another approach would be to investigate the associated differential equations that are needed to obtain the transform constants and focus just on solving them for a simple case.

Isn't this answer mathematica.stackexchange.com/a/178333/9469 completely solves your problem?

@yarchik I have seen your link before your comment. But it can only be mapped from circle to another Region, not between any two Regions

@yode That's the idea. The map you are asking for is a composition of two maps: from the region 1 to a circle and from the circle to the region 2.

@yarchik Circle to the Region maybe be easy, but Region to the Circle is very hard as I know.

Sorry, unfortunately, I have no time for a detailed answer. Fortunately, there a people who have worked much more on this topic. Please have a look into this article by Mark Gillespie, Boris Springborn, and Keenan Crane: dl.acm.org/doi/10.1145/3450626.3459763 and into the references therein. (There is also a video of Mark's talk.) I know, this is not the anwer you were looking for. But it is the best that I can give to you at the moment.

@cvgmt @user64494 can we settle on the title interpunction without SE auto flagging a 'rollback war'?