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