Can you explain why your code does produce a conformal map? I don't understand arguments from the link. TIA.

Actually this is numerical model, therefore it could be conformal in exact arithmetic only. In the real case as above we have some numerical approximation to conformal map. In this example, you can see that only one corner of the triangle is reproduced quite accurately. To reproduce the other 2 angles, you need to increase the number of points in certain places of reg2.

How can we see that it preserves angles?

@AlexTrounnev: Thank you. I still don't understand why "we have some numerical approximation to conformal map". Any picture is not a serious argument.

@user64494 We use f1, f2 to map reg1, reg2 to the unit disk, and inverse f1 to map reg2 to reg1, Theoretically in exact arithmetic it should be conformal map.

@azerbajdzan Do you mean to map mesh to see how it preserves angles?

@Alex Trounev: Yes, something like this.

@AlexTrounnev: Sorry, why is f1 an approximation of a conformal map from reg1 to the unit disk?

@user64494 We use Riemann mapping theorem to compute f1,f2`. Your question probably concerns the corners of the triangle, near which the solution to Laplace's equation should be singular. Is it correct?

@AlexTrounnev: How do we use Riemann mapping theorem to compute f1? Please give us details. No, my request does not concern the corners of the triangle.

@azerbajdzan Please pay attention that in this example we use mesh of 4830 triangle elements. But even this number is not enough to reproduce the border of the triangle reg1 using f2 and inverse f1. If we take a small number of triangular elements then the display will not even look like the desired triangle reg1. Apparently, we can only theoretically assume that angles are preserved with such a map.

@user64494 Do you want me to retell the post that Heinrich Schumacher wrote? :)

@AlexTrounnev: As I wrote I don't understand the argument from the link. As I understand it, you, following the linked answer by Henrik Schumacher, solve the diffusion equation in reg1. Why is it a conformal map from reg1 to the open unit disk?

@user64494 This is standard proof to the Riemann theorem considered in any complex analysis book. See for example Sketch proof via Dirichlet problem on en.wikipedia.org/wiki/Riemann_mapping_theorem

Let us continue this discussion in chat.

It is one-to-one mapping but not conformal mapping as answer by cvgmt shows. Either there is an error in the code or the method is wrong. I believe it is the latter case that the method by @Henrik Schumacher does not produce conformal map in the first place.

@azerbajdzan Maybe you are right. I gonna check how we can improve this model using PI algorithm from the paper doi.org/10.1007/s40315-020-00325-w

@AlexTrounev Good job!

@azerbajdzan You absolutely right that code for vfun has an error. I don't know if it makes sense to look for this error. Since code with PI algorithm implementation looks much simpler.

@cvgmt Thank you very much for your help.