HYPERBOLIC GEOMETRY Paul Klotzle Gabe Richmond

Where does Hyperbolic Geometry Come From?

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.

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

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.

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 .

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.

Time for discovery!!!

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

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)

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