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
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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.
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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.