We could use poincare disc model as a hyperbolic geometry model.
I have difficulty understanding poincare disc model. So is there someone to help?
Asked
Active
Viewed 1,835 times
1
-
1Use an automorphism to place one vertex in the origin. It's pretty evident then. – Daniel Fischer Oct 02 '13 at 16:14
1 Answers
3
You can choose any point within a triangle, and move it to the center of the disk model using a congruence which preserves all angles. Then you have a triangle whose edges are arcs of circles that are orthogonal to the unit circle. As such, they curve inward from the Euclidean connection of the corners. Therefore the hyperbolic angles are smaller than the Euclidean ones.
MvG
- 42,596
-
Actually, it is hard to show a congruence which preserves all angles... Anyway thank you for you answer. – jakeoung Oct 03 '13 at 15:27
-
@topy: Well in the Poincaré model, congruences are those Möbius and Anti-Möbius transformations which preserve the unit disk. A Möbius transformation is conformal, so it preserves angles. – MvG Oct 03 '13 at 15:38
-
Thank you, I will later study that. I've posted new problem. Could you see it? link – jakeoung Oct 03 '13 at 16:04