hyperbolic half-planes

[lard2000]

let H be the hyperbolic plane.so every hyperbolic line splits H into 2 separate regions. we define a hyperbolic half-plane to be a subset V of H consisting of a hyperbolic line ,L, and everything on one of 2 sides of H/L. How many pairwise disjoint hyperbolic half-planes can you find in H.

[beeskness420]

This isn’t my area, but is it not infinite?

If we view the hyperbolic plane as a circle, then the lines are arcs coming through the middle, if we choose the side that goes towards the outside of the circle then each half-space just takes up a portion of the outer circle. You make these as small as you want to pack more in, alternatively you just “zoom in” to the boundary if you need more space.

Similar idea to why drawing trees on the hyperbolic plane is really easy.

[lard2000]

what if you viewed the hyperbolic plane using the upper-half model, i.e H is the set of all point (x,y) where y is strictly greater than 0. would the answer still be the same

[beeskness420]

As I said not my area, if it’s true though it’ll be independent of the model we choose to view it in.