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Homework 4

Due February 21 (Tuesday)
1
Read
: Stahl, Chapters 4.4., 5.1, 5.2, 5.3, 5.4.
1. Find the Euclidean center and radius of the circle that has hyperbolic center (5
,
4) and
radius 3.
2. Prove that the hyperbolic circumference of a circle with hyperbolic radius
R
is 2
π
sinh(
R
).
3. Prove Proposition 5.1.4 in Stahl, that in hyperbolic geometry, every angle is congruent to
an angle in standard position. Recall that in class we showed this for right angles, which
was the hyperbolic analog of Euclid’s Postulate 4.
4. Let
P
be a point in
H
2
, and let
‘
be a Euclidean line segment through
P
. Prove that there
exists a unique hyperbolic geodesic
g
that passes through
P
and is tangent to
‘
. (In other
words, prove that, given a point
P
and a unit vector
v
, there is a unique geodesic through
P
that leaves
P
in the direction of
v
.) Also, give an explicit construction/description of
g
, given
‘
, using either rulerandcompass or analytic methods.
5. Exercise 5.2.6 in Stahl asks you to consider the hyperbolic analog of the Euclidean theorem
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This note was uploaded on 03/11/2012 for the course MTHSC 481 taught by Professor Staff during the Spring '12 term at Clemson.
 Spring '12
 Staff
 Geometry, Topology

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