West Hartford, CT 06117, USA; bDepartment of Mathematics, Eastern Connecticut State University, Willimantic, CT 06226, USA
Abstract: The Poincaré disk model played an important role in the acceptance and development of hyperbolic geometry. Although exceptionally useful, the pedagogical value of the model can be further enhanced via experimentation in a dynamic geometry environment. The focus of this article is on the creation of custom tools in GeoGebra for constructing hyperbolic lines and circles in the Poincaré disk. In an effort to make this material accessible to a wider audience, the necessary mathematics is also included.
Keywords: Poincaré disk, GeoGebra: custom tools
  1. Introduction
Euclid’s Elements was a comprehensive summary of the mathematics known to the ancient Greeks and remained the definitive textbook on geometry for more than 20 centuries. His postulates laid bare many of the assumptions made by the Greek geometers and the fifth postulate, called the parallel postulate, has been of particular interest. All attempts to prove that the fifth postulate was dependent on the first four ended in failure and most of these efforts contributed little to the evolution of geometry. Carl Friedrich Gauss may have been the first mathematician to recognize the independence of the fifth postulate, but he never published his results [7]. Models of hyperbolic geometry came 30 years later and were used to establish the relative consistency of Euclidean and hyperbolic geometry and to prove the independence of the fifth postulate. The Poincaré disk model was originally developed by the Italian mathematician Eugenio Beltrami in 1868 and was reintroduced and popularized by French mathematician Henri Poincaré in 1881. The model is conformal and provides a link to several other branches of mathematics including complex function theory and differential equations. The Poincaré disk model entered the popular culture via the work of the Dutch graphics artist M.C. Escher.
The Poincaré disk and other models made hyperbolic geometry easier to visualize and played a role in its gaining acceptance. Although the models are helpful, students can benefit from the use of dynamic geometry software which provides an environment suitable for experimentation. At present there are several software packages available for this purpose including Cinderella [3] and NonEuclid [2] which both provide tools for working with the Poincaré disk model. These tools are useful, but they are not as well known as GeoGebra [9] and it is natural to wonder whether or not the same tasks could be performed in GeoGebra. Currently, GeoGebra does not provide any built in tools but several custom tools are available on GeoGebraTube. Some of the best include those developed by Maline Christersson [4] and Stehpan Szydlik [10]. However, both of these tools have some defects. For example, their hyperbolic line tools do not work properly in all cases. The emphasis in this paper is on learning how to create robust custom tools in GeoGebra for constructing hyperbolic lines and circles in the Poincaré disk model. In an effort to make this material accessible to a wider audience, the mathematics needed to construct the tools will also be provided.
  1. Poincaré disk model
The Poincaré disk model begins with a circle γ with center O in the Euclidean plane. Points in this model are points in the interior of γ. Points on γ are called ideal points. They are not included in the model and are considered to be infinitely far away from points in the interior. There are two types of lines in the Poincaré disk model. The first type are the open diameters of γ and the second type are the open circular arcs orthogonal to γ. Taken together, the two types of lines are called p-lines and they are geodesics. It is worth noting that a diameter may be regarded as the limiting case of circle whose radius is approaching infinity.