Section 5.2 A Model for Hyperbolic Geometry
Hyperbolic geometry can be drawn with the aid of the Poincaré disc model. The hyperbolic plane is represented by a disc with the border not included ("open disc" in analysis terms). Lines are either diameters or circular arc that are orthogonal to the disc. The origin is the center of the disc. You will use the provided Geogebra file or one you search for online in Geoegebra with special tools to explore hyperbolic geometry.
Checkpoint 5.2.1.
Construct each of the following hyperbolic figures.
(a)
Triangle using at least two lines that are diameters.
(b)
Triangle using exactly one line that is a diameter.
(c)
Triangle using no lines that are diameters.
(d)
Quadrilateral (Can you make it a square?)
Solution.
Here are two versions of a quadrilateral. When the parallel lines are further from the center, the sum of the interior angles of the quadrilateral are less than 360°.
As the parallel lines approach the center of the disk, the sum of the interior angles of the quadrilateral approach 360°.
Checkpoint 5.2.2.
Explore parallelism in hyperbolic geometry.
(a)
Construct a line and select a point not on that line. Construct two lines through that point parallel to the given line.
(b)
How many lines through that point parallel to the given line could be constructed?
Solution.
It appears infinitely many lines parallel to the given line may be constructed.
(c)
Do any of these parallel lines have special properties? Properties might be easier to describe in terms of the model.
Solution.
Opposite the main line, there appears to be two cusps formed by the parallel lines. As well, none of the parallel lines intersect each other more than once.
(d)
Construct two parallel lines. For ease make them large and close in the model. What seems to be true about the distance between the parallel lines?
Solution.It seems like the distance between the parallel lines is constantly changing. They're not equidistant everywhere, but there may (or may not) be distances that are pairwise equidistant as the distance between the lines increases. In some special cases, the parallel lines could appear to be reflections of each other.
