1. Does anyone know if there are any conformal maps of the hyperbolic plane that map to the whole Euclidean plane, or if such a thing is provably impossible?

I know about the disc and half-plane models, but having those areas near the line or the boundary where tiny changes in the Euclidean coordinates make huge (unbounded) differences in the hyperbolic position really throws off any hope of storing object positions in a simulation of a hyperbolic universe.

The other option would be to use a third or fourth coordinate, but I can't figure that out either.

2.

3. Originally Posted by MagiMaster
Does anyone know if there are any conformal maps of the hyperbolic plane that map to the whole Euclidean plane, or if such a thing is provably impossible?

I know about the disc and half-plane models, but having those areas near the line or the boundary where tiny changes in the Euclidean coordinates make huge (unbounded) differences in the hyperbolic position really throws off any hope of storing object positions in a simulation of a hyperbolic universe.

The other option would be to use a third or fourth coordinate, but I can't figure that out either.
If there were sucha map then the hyperbolic plane would be flat. It is not.

4. What about non-conformal maps, or maps that are only conformal at the origin?

5. Originally Posted by MagiMaster
What about non-conformal maps, or maps that are only conformal at the origin?
The hyperbolic plane is homeomorphic to Euclidean space, so yes there would exist non-conformal maps, but you asked about conformal ones.

What are you trying to do ?

6. You can obviously find a map which is conformal inside a small disk. Just map a disk in upper half space to a disk of the same size centered at the origin, and then extend the map continously (albeit non-conformally) to the whole space.

There's a pretty easy way to see that upper half space is not confomally equivalent to the complex plane. For suppose, on the contrary, that the two spaces were conformally equivalent. Let f(z) be any non-constant holomorphic function on the complex plane. Then, if our assumption is true, we can think of f(z) as a map from the plane into upper half space. But upper half space is conformally equivalent to the unit disk. This implies that f(z) is conformally equivalent to a bounded holomorphic function, which would force f(z) to be constant.

7. Good to know.

Basically, I want to make a program (well, any number of programs) that deal with the hyperbolic plane, but storing object coordinates using the Poincare disc has accuracy issues near the border, and since almost everything is near the border...

Anyway, I'm basically looking for ways around that. A map from the Euclidean plane to the hyperbolic plane would allow the use of a much wider range of numbers for object positions. Similarly using an extra coordinate or two to make up for the lost information would work.

I think what I need is some map that bounds the distance between points between the two maps.

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