Mapping points from Euclician 2-space onto a Poincare disc

Question

For some reason it seems that everyone writing webpages about Poincare discs is only concerned with how to represent lines and measure distances.

I'd like to morph a collection of 2D points (as defined by x,y coordinates in the Euclidian plane) onto a Poincare disc, but I have no idea what the algorithm is supposed to be like. At this point I don't even know if it's possible to create a mapping between Euclidian 2-space and a Poincare disc...

Any pointers?

Goodwill, David

Answer 1

If I understand everything correctly, the answer you get on the other forum is for the Beltrami–Klein model. Once you have that, you can get to the coordinates in the Poicare' disk with

p = b / (1 + sqrt(1 - b * b))

Where p is the vector of coordinates in the Poincare' disk (i.e. what you need) and b is the one in the Beltrami–Klein model (i.e. what you get from the other answer).

Answer 2

You describe your data as a collection of points. But from your comments, you want to make lines in the plane still map to lines in the disk. You seem to want to preserve the "structure" of the space somehow, which is probably why you use the term "morph". I think that you want a conformal map.

There is no conformal bijection between the disk and the plane. There is such a mapping between the half-plane and the disk, and it preserves "lines", but not the kind that you want, unfortunately.

You said "I don't even know if it's possible to create a mapping" ... there are a number of mappings for you to choose from (see the Unit Disk page for an example) but there are none with all the features you seem to want.