What are possible interesting panoramic projections of the hyperbolic plane?
I am puzzling with projections of the hyperbolic plane and at the moment specifically with panoramic ones.(my term, maybe there is a better or more frequently used term, maybe that is why I could not find much about it)
I define panoramic projections (again my term, maybe other terms are better) as a projection where there is a dedicated point $O$ and all lines trough point $O$ become parallel vertical lines in the projection.
And a dedicated direction (in this case the positive y axis , that has its image on the $h = 0$ axis. The negative y axis has its image up side down on the $ h=\pm 1 $ lines
So taking the origin as the dedicated point $O$ and a point $A = (x,y)$ where $x$ and $y$ are the coordinates of the point in the Poincare disk model.
the projection of $A$ to $(h,v)$ becomes a modified version of $$ h = \frac{\arctan (x / y) }{\pi} $$ for the horizontal coordinate,so that it goes from $-1$ to $1$
And some function $$ v = f(x,y), (f(x,y) \ge 0 ) $$ for the vertical coordinate.
The result is a projection of the complete hyperbolic plane as if it is seen from the origin, and the viewer there can only look forward and rotate. And the rotation becomes a translation to the left or right in the image. (So all libes through $O$ become parallel lines)
My panoramic projection is comparable to a cylindrical projection in spherical geometry where the dedicated position is a point on the axis , for examples
Mercator projection ( https://en.m.wikipedia.org/wiki/Mercator_projection
Equirectangular projection ( https://en.m.wikipedia.org/wiki/Equirectangular_projection )
and many others, (in each of these examples the designated point is the south pole)
My question is: What are interesting choices for the function $f(x,y) $?
What is the function $f(x,y)$ to make the projection:
- equidistant?
- equal area?
- limited (there is a maximum of v)
- gnomonic (if possible)
- conformal (if possible)
- otherwise interesting?
Maybe i just used the wrong search terms but I could not find a good introductory text on this. If you know one please let me know.