University of Bahrain
College of Science
Department of Mathematics
First Semester 2003/2004
Final Exam
Maths 451
Time: 2 hours
Date: 10/01/2004
Max. Marks: 50
Question I. (15 marks)
A) (3=1+1+1 marks).
Given a geometry ( S , G ) , state what is meant by:
a) Two figures are congruent.
b) An invariant set of figures.
c) An invariant function.
B) (12=4+2+3+3marks)
Let S 2 denote the unit sphere, N  (0,0,1) the north pole, D the unit disc in the (xy)plane and p N denote the stereographic projection of the unit sphere onto the (xy)plane.
2
a) Find a formula for p N and show that it sends the lower hemisphere S  onto the
unit disc D.
b) Let p denote the orthogonal projection onto the xy-plane, that is p(x,y,z)=(x,y).
2
Show that p is a bijection from S  to the unit disc D.
c) Let f : D  D be defined by f ( z )  p N op 1 (z). Find a formula for f and f 1 .
d) Explain how we can get a new model for hyperbolic geometry on the disc by using
the function f. (Such a model is called the Klein model of hyperbolic geometry.)
Question II. (9=3+3+3 marks)
Let H denote the set of Mobius transformations which send the unit disc to itself (the
hyperbolic group).
a) Show that every Euclidean rotation about the origin belongs to H.
i
b) Show that a Euclidean rotation about z  does not belong to H.
2
c) Find a formula for a hyperbolic rotation about z 
i
.
2
Question III. (14=3+3+3+3+2 marks)
Let D denote the unit disc.
a) Prove that through any two distinct points z, w  D passes a unique HSL.
b) Prove that if [zw] is an HSL segment between z and w in D, then it can be
extended to an HSL with infinite hyperbolic length from the side of z and from
the side of w.
c) Given a  D and R>0. Show that there exists a hyperbolic circle in D with
hyperbolic center a and hyperbolic radius R.
d) Let  1 ,  2 be two HSL’s which intersect at right angle at p  D . Show that there
exists a hyperbolic transformation which sends this angle to the standard right
angle at the origin. And therefore deduce that all right angles are congruent.
e) Consequently, hyperbolic geometry satisfies all the first 4 postulates of Euclid.
Show that it does not satisfy the fifth one.
Question IV: [12=6+4+2 marks]
Let C1 and C2 be two clines in the complex plane.
1)
Let Ti  z   z * be the symmetric point of z with respect to the cline C i for i  1,2
(that is Ti is the reflection in C i ).
Prove that T2  T1 is a always a Möbius transformation.
(Hint: discuss all the possible cases where both clines are circles or both are
lines, or … ).
.
2)
If C1 is a hyperbolic straight line in the unit disc. Show that T1 sends the unit
disc to itself, and that T1 is not a mobius transformation. Where as above, T1
denotes the reflection in C1 .
3)
If C1 and C2 are hyperbolic straight lines in D show that T2  T1  H (the
hyperbolic group).