Using the mathematical software The
Geometer’s Sketchpad, we created
the necessary tools to work with the Upper Half-Plane model of
Hyperbolic Geometry in the same way that the Euclidian Geometry. That
is, we made sketches to construct not only the basic ob jects as lines,
segments or rays but circles with a given center and point, center and
radius or the angle bisector as well as the parallelism angle,
perpendicular line, horocycle,... and some of the triangle centers.
All the constructions were made using the environment that Sketchpad
provides. That is, we construct the tools only using synthetic
properties and the
description of the lines and isometries in the Upper Half-Plane model.
For instance, to construct a hyperbolic line that passes through two
given points we use that this line is an euclidian circle perpendicular
to the boundary line. Once we had the basic tools implemented we began
to use them by showing
that the equivalent statements of the fifth Euclidian postulate are not
fulfilled. For example, using the tools it is possible to illustrate
that in the Hyperbolic
plane, given three different points, does not always exist a
circumference meeting the three points. As the Sketchpad makes dynamic
constructions we can plot three any points and then drag them. The
circumference will be drawn only in the cases it exists so that we can
find the existence conditions for the relation between the vertices.
As an example of application we made a second sketch that allow to work
with triangles. Once we know how construct segments we can easily plot
a hyperbolic triangle and then using the hyperbolic bisector,
perpendicular bisector and perpendicular line tools to construct some
of the remarkable points and lines. We also made the isometries of the
Upper Half-Plane and used the tools to construct tessellations. The
isometries allow to see how the Euclidian and Hyperbolic length differ
but the angles do not.
The tools are of public domain although the program The Geometer
Sketchpad is not. They can be found in the web page of the The Geometer
Sketchpad,
http://www.keypress.com/sketchpad/general
resources/advanced sketch gallery/index.php
in the topic Beyond Euclid under the title Half Plane Model of
Hyperbolic Geometry and also in my homepage (
http://mat.uab.cat/~juditab/HypGeom.htm)
where it is explained step by step how they are constructed. These
tools can be useful to study and see the differences between Euclidean
and Hyperbolic Geometry in an easier and interactive way so that they
are appropriated to be used in the beginning of the study of the
Hyperbolic Geometry. These tools make a class more dynamic in the sense
that the teacher can make constructions and demonstrations with the
program in an easy and quick way without having to draw again the
picture to explain a similar but different case. Moreover, students can
state hypothesis that the can be verified instantaneously in a visual
way and, if necessary, make changes without having to clean the
picture. However, they can not only be considered as mathematical
education software applied in Geometry but they can be used in other
areas as well specially in the Analysis area.