Ideas in Geometry/Hyperbolic Geometry
What is Hyperbolic Geometry? - Hyperbolic geometry allows us to follow Euclid's first four axioms then replacing the last one: Parallel Postulate. What are the axioms that hold true in this geometry then? 1. A line can be drawn from a point to any other point. 2. A finite line can e extended indefinately. 3. A circle can be drawn, given a center and a radius. 4. All right angles are 90 degrees. 5. Given a line and a point not on that line there exists more than one parallel to the given line through the given point. What does this mean? In hyperbolic geometry the Poincaré Half-Plane Model we work in the upper half-plane (x,y) when x > 0. - here points are still points the difference is that lines are actually arcs inside the boundary where the intersection point with the boundary forms a right angle. This causes a small distortion around the boundary causing these distances to look much smaller and closer together than they truely are when near the bourndary. The closer an object is to a boundary the larger it is with more area it can hold inside of it. This also means that the shortest distance between two points on a line lies between the two points the farthest from the boundary line.