The Hyperbolic Toolbox: 
Non-Euclidean Constructions in Geometer's Sketchpad

6.  Teaching Examples

• Have students develop some of the tools themselves.  The difficulty of writing the Sketchpad scripts for the basic "hyperbolic straightedge and compass" constructions varies greatly.  In the Klein model, for example, constructing lines and segments are trivial, calculating lengths, dropping and raising perpendiculars is relatively straightforward, while calculating angle measure, constructing midpoints, angle bisectors, and circles is quite challenging.

Interestingly, the corresponding constructions in the other models do not necessarily have corresponding difficulty levels.  For example, writing a tool that constructs a hyperbolic line in the Klein model is merely a matter of drawing a chord passing through two points interior to a circle.  Constructing a half-circle passing through two points in the Poincaré half-plane is slightly more challenging, though students who are able to circumscribe a circle about a triangle should be able to complete this task.  Constructing a hyperbolic line in the Poincaré disk requires several steps, including the ability to find "inverse points" (see Greenberg p.243-247).  On the other hand, calculating angle measure in the Poincaré models is relatively straightforward since these models are conformal, while in the Klein model, this action requires the use of hyperbolic trigonometry.

In my classroom, I have had students develop the tools for constructing lines and segments in the three models as an introduction to the models and to writing scripts in Geometer's Sketchpad after reading the appropriate section of  Greenberg for the Poincaré disk construction.  In addition, I have the students discover on their own and without a computer how to "raise perpendiculars" in the Poincaré models.  For more advanced classes, the more challenging constructions could  be assigned as individual or group projects.
 

• Use the scripts as a toolbox for more extensive hyperbolic constructions.  The scripts can serve as building blocks for Geometer's Sketchpad activities that lead to a more complete conceptualization of hyperbolic geometry.  The power of the tools becomes most apparent here:  students are able to investigate the subject interactively as they develop their own constructions.  Moreover, constructions require justification.  After completing a construction, the natural question to ask is, "Is it correct?"  Here are some samples that illustrate these principles:
  • Example 6.1:  What Can You Build?
  • Example 6.2:  "Proving the Parallel Postulate"
  • Example 6.3:  The Angle of Parallelism
     
     
• Use the scripts to either motivate or help students visualize major theorems in hyperbolic geometry.  Many of the theorems of hyperbolic geometry are counterintuitive for students who have until now considered only Euclid's geometry.  For these students, the hyperbolic models can be a welcome relief.  Abstract theorems can give way to illuminating examples, illustrating many of the properties of non-Euclidean geometry.


Although not truly applications in themselves, the hyperbolic tools can be used (by either the instructor or students) to develop applets illustrating hyperbolic geometry.

• Example 6.4: Common Perpendiculars
• Example 6.5: An Absolute Standard of Length