# Are there other models for 2 dimensional hyperbolic geometry?

I was a bit browsing the internet for models for (2-dimensional) hyperbolic geometry.

and realised that besides the well known

• Poincare half plane model
• Poincare disk model
• Beltrami-Klein disk model
• Hyperboliod ( Weierstrass- Minkowski- Lorentz- ) model

There are also the

• Hemisphere model (mostly used for transformations between the Poincare half plane model and the other models)

And via via I came in contact with the

• Gans Model (flattened hyperboloid model, a hyperboloid model minus the z coordinate) (see Gans, David . A New Model of the Hyperbolic Plane. American Mathematical Monthly, Vol. 73, Issue 3, March 1966. or www.d.umn.edu/cs/thesis/kedar_bhumkar_ms.pdf‎ )

This made me wonder: are there even more models of (2-dimensional) hyperbolic geometry that I should know but haven't heard of? (do add references)

• You got the main models... I don't think I've run across any radically different ones yet. – rschwieb May 5 '14 at 18:10

## 1 Answer

I believe Hubbard uses a 'belt model' in Teichmuller Theory: Volume I, but I don't have a copy on me at the moment.

Just for the sake of completeness --- in principle, by the Riemann mapping theorem, any simply connected domain in $\mathbb{C}$ can serve as a model of hyperbolic space.

Edit: I have recently learned of a parabolic model, defined by intersecting a vertical affine plane in $\mathbb{R}^{2+1}$ with the forward light cone and projectivizing. By the same method, any conic section should define a model of $\mathbb{H}^2$.

• A generalization of Hubbard's "belt model" would be to take any k-dimensional totally geodesic hyperbolic subspace of hyperbolic n-dimensional space and on the normal bundle pull-back the metric via the exponential map. – Ryan Budney May 24 '14 at 22:22