# Hyperbolic geometry

1. ### kexue

196
I like to learn some basic hyperbolic geometry!

Starting with the hyperbolic plane, the upper half plane with the hyperbolic lines being all half-lines perpendicular to the x-axis, together with all semi-circles with center on the x-axis.

Why and how are there always infinitely many hyperbolic lines parallel to a given hyperbolic line L and passing through a given point p not lying on L?

Does parallel in hyperbolic geometry just mean two lines do not cut each other?

Second problem.

My book says the boundary of the hyperbolic plane can be thought of as real line $$\cup$$ infinity, which is supposed to be a circle. I can't see this to be a circle. What does the author mean?

thank you

626
3. ### wofsy

707
In my opinion hyperbolic geometry is best studied at first on the unit disc where classical Moebuis tranformations of the complex plane that preserve the disc are isometries. It is immediate from this way of looking at it that there are infinitely many parallels to a line through any point.

For the upper half plane, it is good to map it onto the disc conformally and take the induced metric.

4. ### wofsy

707
On the unit disc the boudary is the unti circle, no problem here. This circle is mapped to the real line plus the point at infinity on the Riemann sphere by a conformal mapping of the unit disc onto the upper half plane. I urge you to learn it this way.