How and for that

During the last year of my degree, one of the optional subjects that I could study was "A directed work" where the student can choose the contents of the subject to study a theme he is interested in. One of the works, proposed by the professor Eduardo Gallego, consisted on studying the plane Hyperbolic Geometry from an axiomatic point of view to create tools with Sketchpad in order to be able to work with the Poincare's Half-Plane model in the same way as it can be done with the Geometry Euclidian. The subject seemed very interesting for me since I have always been interested in Geometry and I thought that was a good opportunity to learn about Hyperbolic Geometry.

It seemed that with Sketchpad we could attain the goal of the work since it allows to create macros with the tools that the user creates. So, it was only necessary to think which are the basic objects to work in a geometry. We have two types of objects, the basics ones, from which we can create all the other ones, and those we obtain from the basic ones. According to the pattern in the Geometry with rule and compasses we have got that the basic constructions are: straight line, ray, segment, circumference given the center and a point, given the center and a radius. From these we can create any of the other tools, such as, the perpendicular straight line, the angle bisector, the perpendicular bisector, the parallel straight lines to a fixed straight line, the equidistant or the horocycle. All these were constructed from the basic tools, that are already created in Sketchpad so we only had to select the ones we need to use. You will be able to find these tools in Tools to work with the Hyperbolic Geometry (file .gsp to download). Also, to be able to give the construction of each one of the former tools it was first necessary to know the properties of these objects and how they are represented in the Half-Plane Model. Because of that we did first a theoretical part where we studied these properties at the same time that the basic characteristics of the Hyperbolic Geometry (Workshop in Hyperbolic Geometry.pdf, in catalan).

Once the tools were constructed we wanted to use them to study the triangles. For example, some of the differences that we had studied in the theoretical part it was that in the Hyperbolic Geometry, the sum of the angles of a triangle is always less than two right angles. With the tool of angles measure we could check out in a fast way and with many examples that this is certain. We can also check that the circumcenter or the orthocenter not always exist. These examples and the construction of some other remarkable points and lines are in Some examples.