1. Hyperbolic Geometry
Chapter 9

2. 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?

3. Poincaré Disk Model Model of geometric world Rules
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

4. Poincaré Disk Model Euclid’s first four postulates hold
Given two distinct points, A and B,  a unique line passing through them
Any line segment can be extended indefinitely
A segment has end points (closed)
Given two distinct points, A and B, a circle with radius AB can be drawn
Any two right angles are congruent

5. Hyperbolic Triangles
Recall Activity 2 – so … how do you find  measure?
We find sum of angles might not be 180

6. Hyperbolic Triangles Lines that do not intersect are parallel lines
What if a triangle could have 3 vertices on the fundamental circle?

7. Hyperbolic Triangles Note the angle measurements
What can you conclude when an angle is 0 ?

8. 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

9. Hyperbolic Polygons
What does the hyperbolic plane do to the sum of the measures of angles of polygons?

10. What seems “wrong” with these results?
Hyperbolic Circles
A circle is the locus of points equidistant from a fixed point, the center
Recall Activity 9.5
What seems “wrong” with these results?

11. 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

12. Distance on Poincarè Disk Model
Rule for measuring distance  metric
Euclidian distance
Metric Axioms
d(A, B) = 0  A = B
d(A, B) = d(B, A)
Given A, B, C points, d(A, B) + d(B, C)  d(A, C)

13. Distance on Poincarè Disk Model
Formula for distance
Where AM, AN, BN, BM are Euclidian distances M N

14. Distance on Poincarè Disk Model
Now work through axioms
d(A, B) = 0  A = B
d(A, B) = d(B, A)
Given A, B, C points, d(A, B) + d(B, C)  d(A, C)

15. Circumcircles, Incircles of Hyperbolic Triangles
Consider Activity 9.3a
Concurrency of perpendicular bisectors

16. Circumcircles, Incircles of Hyperbolic Triangles
Consider Activity 9.3b
Circumcircle

17. 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

18. Circumcircles, Incircles of Hyperbolic Triangles
Note issue of  bisectors sometimes not intersecting
More on this later …

19. Circumcircles, Incircles of Hyperbolic Triangles
Recall Activity 9.4
Concurrence of angle bisectors

20. Circumcircles, Incircles of Hyperbolic Triangles
Recall Activity 9.4
Resulting incenter

21. 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

22. 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!!

23. Parallel Postulate in Poincaré Disk
Playfair’s Postulate
Given any line l and any point P not on l,  exactly one line on P that is parallel to l
Definition 9.4 Two lines, l and m are parallel if the do not intersect l P

24. 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?
No lines through P, parallel
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

25. 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

26. Elliptic Parallel Postulate
Flat map with great circle will often be a distorted “straight” line

27. 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

28. 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

29. Hyperbolic Parallel Postulate
Result of hyperbolic parallel postulate Theorem 9.4
There is at least one triangle whose angle sum is less than the sum of two right angles

30. Hyperbolic Parallel Postulate
Proof:
We know  at least two lines parallel to l
Note  to l, PQ
Also  to PQ, m and thus || to l
Note line n also || to l

31. Hyperbolic Parallel Postulate
XPY > 0
Not R on l such that we have PQR
QPR < QPY
Move R towards fundamental circle, QRP  0
Thus QRP < XPY
And PQR has one rt. angle and the other two sum < 90
Thus sum of angles < 180

32. Parallel Lines, Hyperbolic Plane
Theorem 9.5 Hyperbolic Parallel Theorem Given any line l, any point P, not on l,  at leas two lines through P, parallel to l
Remember parallel means they don’t intersect

33. 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 

34. Parallel Lines, Hyperbolic Plane
Consider Activity 9.7
Note the congruent angles, DCE  FCD

35. Parallel Lines, Hyperbolic Plane
Angles DCE & FCD are called the angles of parallelism
The angle between one of the limiting rays and CD
Theorem 9.6 The two angles of parallelism are congruent

36. Parallel Lines, Hyperbolic Plane
Note results of Activity 9.8
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

37. Quadrilaterals, Hyperbolic Plane
Recall results of Activity 9.9
90 angles at B and A

38. Quadrilaterals, Hyperbolic Plane
Recall results of Activity 9.10
90 angles at B, A, and D only
Called a Lambert quadrilateral

39. Quadrilaterals, Hyperbolic Plane
Saccheri quadrilateral
A pair of congruent sides
Both perpendicular to a third side

40. 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)

41. Hyperbolic Geometry
Chapter 9