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Hyperbolic Geometry
Chapter 11
Hyperbolic Lines and Segments
• Poincaré disk model
 Line = circular arc, meets fundamental circle
orthogonally
• Note:
 Lines closer to
center of fundamental
circle are closer to
Euclidian lines
 Why?
Poincaré Disk Model
• Model of geometric world
 Different set of rules apply
• Rules
 Points are interior to fundamental circle
 Lines are circular arcs orthogonal to
fundamental circle
 Points where line meets fundamental circle
are ideal points -- this set called 
• Can be thought of as “infinity” in this context
Poincaré Disk Model
Euclid’s first four postulates hold
1.Given two distinct points, A and B,  a
unique line passing through them
2.Any line segment can be extended
indefinitely

A segment has end points (closed)
3.Given two distinct points, A and B, a circle
with radius AB can be drawn
4.Any two right angles are congruent
Hyperbolic Triangles
• Recall Activity 2 – so … how do you find 
measure?
• We find sum of angles might not be 180
Hyperbolic Triangles
• Lines that do not intersect are parallel lines
• What if a triangle could have 3 vertices on
the fundamental circle?
Hyperbolic Triangles
• Note the angle measurements
• What can you conclude
when an angle is 0 ?
Hyperbolic Triangles
• Generally the sum of the angles of a
hyperbolic triangle is less than 180
• The difference between the calculated
sum and 180 is called the defect of the
triangle
• Calculate
the defect
Hyperbolic Polygons
• What does the hyperbolic plane do to the
sum of the
measures
of angles
of polygons?
Hyperbolic Circles
• A circle is the locus of points equidistant
from a fixed point, the center
What
seems
“wrong”
with these
results?
• Recall Activity 11.2
Hyperbolic Circles
• What happens
when the center
or a point on the
circle approaches
“infinity”?
• If center could be
on fundamental
circle
 “Infinite” radius
 Called a horocycle
Distance on Poincarè Disk Model
• Rule for measuring distance  metric
• Euclidian distance
d  A, B  
 a1  b1    a2  b2 
Metric Axioms
1.d(A, B) = 0  A = B
2.d(A, B) = d(B, A)
3.Given A, B, C points,
d(A, B) + d(B, C)  d(A, C)
2
2
Distance on Poincarè Disk Model
AM / AN
AM  BN
• Formula for d ( A, B)  ln
 ln
BM / BN
AN  BM
distance
M
N
 Where AM, AN, BN, BM are Euclidian
distances
Distance on Poincarè Disk Model
AM / AN
AM  BN
d ( A, B)  ln
 ln
BM / BN
AN  BM
Now work through axioms
1.d(A, B) = 0  A = B
2.d(A, B) = d(B, A)
3.Given A, B, C points,
d(A, B) + d(B, C)  d(A, C)
Circumcircles, Incircles
of Hyperbolic Triangles
• Consider Activity 11.6a
 Concurrency of perpendicular bisectors
Circumcircles, Incircles
of Hyperbolic Triangles
• Consider Activity 11.6b
 Circumcircle
Circumcircles, Incircles
of Hyperbolic Triangles
• Conjecture
 Three perpendicular bisectors of sides of
Poincarè disk are concurrent at O
 Circle with center O, radius OA also contains
points B and C
Circumcircles, Incircles
of Hyperbolic Triangles
• Note issue of  bisectors sometimes not
intersecting
 More on this later …
Circumcircles, Incircles
of Hyperbolic Triangles
• Recall Activity 11.7
 Concurrence of angle bisectors
Circumcircles, Incircles
of Hyperbolic Triangles
• Recall Activity 11.7
 Resulting incenter
Circumcircles, Incircles
of Hyperbolic Triangles
• Conjecture
 Three angle bisectors of sides of Poincarè
disk are concurrent at O
 Circle with center O, radius tangent to one
side is tangent to all three sides
Congruence of Triangles in
Hyperbolic Plane
• Visual inspection unreliable
• Must use axioms, theorems of hyperbolic
plane
 First four axioms are available
• We will find that AAA is now a valid
criterion for congruent triangles!!
Parallel Postulate in
Poincaré Disk
• Playfair’s Postulate
l
l
 Given any line and any point P not on ,  exactly
one line on P that is parallel to l
• Definition 11.4
Two lines, l and m are parallel if the do not
intersect
P
l
Parallel Postulate in
Poincaré Disk
• Playfare’s postulate Says  exactly one
line through point P, parallel to line
• What are two possible negations to the
postulate?
1. No lines through P, parallel
2. Many lines through P, parallel
Restate the first – Elliptic Parallel Postulate
 There is a line l and a point P not on l such
that every line through P intersects l
Elliptic Parallel Postulate
• Examples of elliptic space
 Spherical geometry
• Great circle
 “Straight” line on the sphere
 Part of a circle with center at
center of sphere
Elliptic Parallel Postulate
• Flat map with great circle will often be a
distorted “straight” line
Elliptic Parallel Postulate
• Elliptic Parallel Theorem
 Given any line l and a point P not on l every
line through P intersects l
• Let line l be the equator
 All other lines (great
circles) through any point
must intersect the equator
Hyperbolic Parallel Postulate
• Hyperbolic Parallel Postulate
 There is a line l and a point P not on l such
that …
more than one line through P is parallel to l
Parallel Lines, Hyperbolic Plane
• Lines outside the
limiting rays will be
parallel to line AB
 Called
ultraparallel or
superparallel or
hyperparallel
 Note line ED is limiting parallel with D at 
Parallel Lines, Hyperbolic Plane
• Consider Activity 11.8
 Note the congruent angles, DCE  FCD
Parallel Lines, Hyperbolic Plane
• Angles DCE & FCD are called the
angles of parallelism
 The angle between
one of the limiting
rays and CD
• Theorem 11.4
The two angles
of parallelism
are congruent
Hyperbolic Parallel Postulate
• Result of hyperbolic parallel postulate
Theorem 11.4
 For a given line l and a point P not on l, the
two angles of parallelism are congruent
• Theorem 11.5
 For a given line l and a point P not on l, the
two angles of parallelism are acute
The Exterior Angle Theorem
• Theorem 11.6
 If ABC is a triangle in the hyperbolic plane
and BCD is exterior for this triangle, then
 BCD is larger than either  CAB or  ABC.
Parallel Lines, Hyperbolic Plane
• Note results of Activity 11.8
 CD is a common
perpendicular to
lines AB, HF
• Can be proved in
this context
 If two lines do not
intersect then either
they are limiting parallels
or have a common
perpendicular
Quadrilaterals, Hyperbolic Plane
• Recall results of Activity 11.9
• 90 angles at B and A`
Quadrilaterals, Hyperbolic Plane
• Recall results of Activity 11.10
• 90 angles at B, A, and D only
• Called a Lambert quadrilateral
Quadrilaterals, Hyperbolic Plane
• Saccheri quadrilateral
 A pair of congruent sides
 Both perpendicular to a third side
Quadrilaterals, Hyperbolic Plane
• Angles at A and B are base angles
• Angles at E and F are
summit angles
 Note they are congruent
• Side EF is the summit
• You should have found
not possible to construct
rectangle (4 right angles)
Hyperbolic Geometry
Chapter 11