Constructions in the Hyperbolic Models
Below are dynamic constructions illustrating several examples from my article "Drawing the Line: Constructions in Hyperbolic Geometry," published in the GeT News [insert reference]. The examples demonstrate various aspects of constructions in three models of hyperbolic geometry: the Klein disk, the Poincare disk, and the Poincaré half-plane. All the examples are created using GeoGebra and the tools found on my hyperbolic tools page:
- Example 1: Visualizing the hyperbolic axiom in the hyperbolic models. In each case, we can see that given the line l and the point P not on l, there are (at least) two lines m and m' through P parallel to l.
- The hyperbolic axiom in the Klein disk.
- The hyperbolic axiom in the Poincaré disk.
- The hyperbolic axiom in the Poincaré half-plane.
- The hyperbolic axiom in the Klein disk.
- Example 2: Constructing a hyperbolic line in the Poincaré half-plane.
- Example 3: Constructing a hyperbolic line in the Poincaré disk.
- Example 4: Finding the flaw in an attempted proof of the parallel postulate in neutral geometry using the Klein disk. By moving the various free points appropriately, one can identify the flaw by seeing where the construction falls apart.