Homework 5

Due March 6 (Tuesday)
1
Read
: Stahl Chapters 6.1, 6.2, 7.1, 7.2, 8.1, 8.2, 8.3.
1. Prove that if 0
< α <
2
π
then there is a hyperbolic quadrilateral whose angles sum to
α
.
2. For
a,b >
0, let
R
(
a,b
) be the Euclidean rectangle with corners (0
,
2), (
a,
2), (0
,
2 +
b
),
and (
a,
2 +
b
).
(a) Give a qualitative description of what
R
looks like to a resident of the hyperbolic
plane. Speciﬁcally, which of its sides are hyperbolically straight or hyperbolically
curved. If curved, in which direction?
(b) For a given
a,b >
0, let ha(
R
(
a,b
)) denote the hyperbolic area of
R
(
a,b
), let
L
(
a,b
)
be the hyperbolic length of the left side of
R
, and let
B
(
a,b
) be the hyperbolic length
of the bottom side of
R
. Prove that
lim
a,b
→
0
+
L
(
a,b
)
B
(
a,b
)
ha(
R
(
a,b
))
= 1
.
For the next two problems, consider a hyperbolic right triangle
ABC
with right angle at
C
, let
α
and
β
denote the angles at
A
and
B
, respectively, and let
a
,
b
, and
c
be the
hyperbolic lengths of the sides opposite