CHAPTER 4
THE HYPERBOLIC PLANE
Definitions and Propositions:
1. The upper half plane is referred to as the hyperbolic plane or the Poincare upper half
plane.
2. The hyperbolic length of an arbitrary curve C is the expression

C
dx 2  dy 2
.
y
Example: The hyperbolic length of the segment of the parabola y  x 2 that joins the
1  4x 2
2 x 2 dx .
3. Proposition: Let q be a circle with center C(c, 0) and radius r. If P and Q are points of q
such that the radii CP and CQ make angles x and y (x < y) respectively, with the positive
csc y  cot y
x-axis, then the hyperbolic length of arc PQ = ln
csc x  csc x
[Hint: x  c  r cos t , y  r sin t ]
3
points (2, 4) to (3, 9) is
4. Corollary: If PQ and RS are arcs of Euclidean circles which share the same center C
on the x-axis and such that each of the triples CPR and CQS is collinear, then the two
arcs PQ and RS have the same hyperbolic length.
5. Proposition: The hyperbolic length of the Euclidean line segment joining the points
P(a, b) and Q(a, c), 0  b  c, is ln( c / b)
6. Show by example that in the hyperbolic plane, the Euclidean segment joining two points
does not provide the shortest distance between those two points.
7. Theorem: (Hyperbolic Straight Lines) The geodesic segments of the hyperbolic
plane are either
 Arcs of Euclidean semicircles that are centered on the x-axis or
 Segments of Euclidean straight lines perpendicular to the x-axis.
8. Exercise: Find the hyperbolic length of the segment joining (8, 4) and (0, 8).
9. Exercise: Find the hyperbolic midpoint of the geodesic segment joining (8, 4) and (0, 8).
10. Hyperbolic Angles: The hyperbolic measure of an angle is identical to the Euclidean
measure.
11. Hyperbolic Rigid Motions:
Theorem: The following transformations of the hyperbolic plane preserve both
hyperbolic lengths measures of angles.
a. Inversions I C , r where C is on the x-axis
b. Reflections rm , where m is perpendicular to the x-axis
c. Translations TAB where AB is parallel to the x-axis.
12. Corollary: All transformations listed in theorem above carry geodesics to geodesics.
Riemannian Geometry
The essential difference between Euclidean Geometry and Hyperbolic Geometry is that same
curves have different lengths by the definition of length in each geometry. In Riemannian
Geometry the length of an arbitrary curve C lying in the plane is given by the expression

Edx 2  2 Fdxdy  Gdy 2 ,
… (A)
C
where E, F, and G are functions of x and y satisfying
E  0, G  0, EG  F 2  0 (provides for a positive length)
A closer look at the integrand in the integral above.
Let C : x  x(t ),
y  y (t ), a  t  b . Then
b

Edx  2 Fdxdy  Gdy  
C
a
When E  G 
2
2
2
2
 dx 
 dx  dy 
 dy 
E   2 F     G  dt
 dt 
 dt  dt 
 dt 
1
, &F 0
y2
(A) reduces to the hyperbolic length.
1
, &F 0
y2
When E  G  1, & F  0 , we get the Euclidean length of the curve parametrized by
C : x  x, y  f ( x), a  x  b .
When E  G 
Note:

Edx 2  2 Fdxdy  Gdy 2 is called the Riemannian metric

dx 2  dy 2 is called the Euclidean metric, &

dx 2  dy 2
is called the Poincare metric
y
Exercises (In the exercises below, h- refers to hyperbolic):
1. Compute the h-length of the curve joining (0, 1) and (1, 2) on y  x 2  1 .
2. Compute the h-length of the arc of the circle x 2  y 2  25, y  0 that lies between (3, 4)
and (-4, 3).
3. Find the point that divides the geodesic segment (0, 10) and (0, 1) such that the ratio
of the h-points is 3:2.
4. Find the h-distance between each pair of the three points (-1, 2), (3, 2) and (0, 1) and
compare them to the corresponding Euclidean distances
5. Find a hyperbolic rigid motion that transforms the point (0, 1) to the point (0, 2)
6. Find a h-rigid motion that transforms the bowed geodesic from (0, 0) to (4, 0) to the
straight geodesic x  10
7. Find a h-rigid motion that transforms the bowed geodesic from (- 4, 0) to (2, 0) to the
bowed geodesic from (10, 0) to (20, 0).
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Proof Writing (due Nov 20th, 2008):
8. Let A and B be two points joined by a bowed geodesic G. Let C(c, 0) be a point on the
x-axis and M(c, y) be the point of G above C. Then, M is the h-midpoint of G if
BCM  ACM .
9. Show that if the points P( x1 , y1 ) & Q( x2 , y2 ) are contained in a bowed geodesic with
( x1  c  r ) y 2
y1 ( x2  c  r )
10. Describe a Euclidean method for constructing some h-isosceles triangle
11. Given a h-geodesic segment, describe a Euclidean method for constructing its
perpendicular bisector.
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Euclidean center (c, 0), and radius r, then h( P, Q)  ln
12. Consider the Riemann metric ydx 2  2 xdxdy  x 2 dy 2 .
EG  F 2  x 2 ( y  1)  0 for x  0, y  1 . Compute the length of three curves joining
(2, 2) and (4,4) relative to this metric.
 The first curve is that joining (2, 2) and (4, 2) and then joining (4, 2) to (4, 4);
 The second curve is joining (2, 2) and (2, 4) and then joining (2, 4) to (4, 4);
 The third curve is the straight segment joining (2, 2) and (4, 4).
HONORS: Exercises 1 – 7, section 4.5