1. HYPERBOLIC GEOMETRY
Paul Klotzle
Gabe Richmond

2. Where does Hyperbolic Geometry Come From?

3. Euclid’s Parallel Postulate
If l is any line and P any point not on l, there exists in the plane of l and P one and only one line m that passes through P and is parallel to l.
Hyperbolic
Parallel Postulate
Spherical
Parallel Postulate
If l is any line and P any point not on l, there exists more than one line passing through P parallel to l.
If l is any line and P any point not on l, there are no lines through P that are parallel to l.

4. History of Hyperbolic Geometry
Famous Mathematics: Saccheri, Gauss, Bolyai and Lobachevski (published in 1800’s)
Met with great criticism
Discovered trying to prove Parallel Postulate
Einstein’ Theory of General Relativity
Currently used to predict orbit of objects in gradational fields, space travel and astronomy

5. Hyperbolic geometry at work
Deflected image
Light ray from distant star
Actual location
Location of star is distorted and shortest path is a curved line.

6. How does Hyperbolic Geometry Work?
Axioms in Hyperbolic Geometry:
Two Points determine a line
A straight line can be extended without limitation
Given a point and a distance a circle can be drawn with the point as center and the distance as radius
All right angles are equal
The Parallel Postulate: If  l is any line and P is any point not on l , then there exists at least two lines through P that are parallel to l .

7. Poincare’s Disk Model
Interior region of a disk, excluding the boundary
Distance from center to boundary is infinite
All lines are arcs of a circle orthogonal to the boundary
All points in the interior of the circle are part of the hyperbolic plane
Angle measures are determined by tangent lines draw at the point of intersection to the arcs.

8. Time for discovery!!!

9. What Euclidean Principals Hold up in Hyperbolic Geometry?
SAS, SSS, and ASA all hold true for triangle congruence
Isosceles Triangle theorem
Basic centers of circles (circumcenter, etc.)
Betweenness holds in hyperbolic
All proofs in Euclidean geometry that do not use parallel postulate
AAA holds true for congruency instead of similarity

10. What Euclidean Principles do not hold up in Hyperbolic Geometry?
Parallel lines are not equal distance from each other
Corresponding and alternating interior angles are not congruent
Euclidean distance does not hold, distance is measured from center of circle
Angle sum of a triangle is less than 180◦
Given a line l and a point A not on the line, there is not a unique parallel line to l through A.
a² + b² < c², area for a triangle is ∏ - (m<A + m<B + m<C)

11. References
Kay, David. (2001). College Geometry – A Discovery Approach. Boston: Addison-Wesley