Math 455 Hyperbolic Geometry
These resemble questions you will see on the final exam.
Name_________________________
1. Find the equation of the half circle through the points 4+3i and -1+2i in the Poincare Half Plane.
2. Where are the following figures “sent” by the PC function? Show calculations and draw graphs to show your work. a) The horizontal line through 5i on the upper half plane. b) The horizontal line through -5i on the lower half plane. c) The vertical line through the origin. d) The vertical line through x = 5 e) The vertical line through x = -5 f) The half circle of radius 5 centered at the origin.

Math 455 Hyperbolic Geometry Name_________________________
3. Where are the following figures sent by the PUC function (the inverse of the PC function? Show calculations and draw pictures. a) The unit circle b) The circle of radius 10 c) The circle of radius 1/10 d) The real axis. e) The imaginary axis
4. Only horizontal translations are possible on the upper half plane. Why do you think this is so? Give an example of an LFT that is a translation of the half plane into itself.
5. Use the Poincare distance function to find the hyperbolic distance on the upper half plane for the following a) the vertical distance between 2i an 10i b) the distance on a half-circle of radius 5 centered at the origin between -45 o
and 45 o
. c) Any complete half circle d) Any complete half line.

Math 455 Hyperbolic Geometry Name_________________________
The upper half plane can be mapped conformally to the unit disk with the
Möbius transformation where w is the point on the unit disk that corresponds to the point z in the upper half plane. In this mapping, the constant z
0
can be any point in the upper half plane; it will be mapped to the center of the disk. The real axis maps to the edge of the unit disk | w | = 1. The constant real number φ can be used to rotate the disk by an arbitrary fixed amount.
The canonical mapping is which takes i to the center of the disk, and 0 to the bottom of the disk.