The founders of a "new geometry"are Gauss, Janos Bolyai and Nikolai Lobachevski for they were the first to deny the absolute nature of Euclidean geometry and provide us with a theory based on axiomatic methods.
remains to finish the story is a particular
realization of non Euclidean
The main problem in the visualization of the Hyperbolic plane is our lack of experience about it. We solve it using various models of the Hyperbolic plane. A model is an assignment of mathematical objects from Euclidean geometry to "play the role of" the non-Euclidean objects.
We visualize the
following models of the hyperbolic plane using our
Our Mathematica® package provides you with the possibility to choose between them.
Entire, infinite two-dimensional Hyperbolic Space is realized in the interior of the unit disk or in the upper halfplane, depending on the model. Boundary circle or the ordinate represent the infinity, so parallel are those lines whose intersection points lie in "infinity".
The lines drawn in the models are not Euclidean lines. Moreover, their look depend on the model. Here is a table showing what are the lines in different models:
The following picture shows a 'square" in these models; exactly the same polygon obtained from one by isomorphisms between models:
All pictures so far are produced using our package. It contains implementation of all basic objects - points, lines, polygons, circles...
Here is a same triangle triangle in all three models.
An important feature of our package is that it provides various motions (isometries) of the plane. The isometries are implemented as a compositions of symmetries and can be easily applied even to the most complicated objects. We start by rotation of our polygon around a given point:
Now, we reflect it with respect to a given line:
"translate" it from one point to another:
And finally, rotate it around an infinite point - this isometry is called horocyclic rotation.