0201 The model: the Poincaré model of hyperbolic geometry
Given a set of geometric properties, we call a model of the set any realization of this set, abstract or concrete, with the same properties as those in the initial set. For instance, if a circle on a piece of paper or on a blackboard, on the screen or in the sand is drawn, all of these are models of the same geometric object, namely the circle.
In our investigations about hyperbolic geometry, the situation is somewhat more complex. We will create a model of the hyperbolic plane on a disk that will be displayed in the Graphics View of GeoGebra, and the graphical output will be shown on the screen or on a piece of paper. To be unambiguous, if required, we will write the letter H before the name of the modeled concept if it is actually a hyperbolic — or sometimes also an absolute — geometry object.
| The modeled concept: | The model (in the Euclidean plane): |
| H-plane | Disk |
| H-point | Inner point of the disk |
| H-line | An arc of a circle which is perpendicular to the boundary of the disk, the arc is inside the disk |
| H-segment | An arc between two points of an H-line |
| H-reflection | An inversion with respect to an H-line or H-segment |
This model of the hyperbolic geometry will be called P-model after its creator Henri Poincaré (1854-1912).
By using dynamic geometry the entire H-line can be displayed, including the border of the disk.
This opens up new horizons in abstraction. As already mentioned, Bolyai proved that Euclidean geometry is a limiting case of the hyperbolic geometry system. If the disk of the P-model is much larger than the visible construction we work with, we can obtain drawings similar to the Euclidean ones, by using the P-model. On a much-enlarged disk it is hard to notice that we work in the P-model, just as with standing on the floor it is hard to notice that we live on a spherical planet.
The power of the P-model to influence one’s understanding
Let us consider a drawing in the Euclidean plane, assumed it is drawn in school. It is a model that shows a part of the Euclidean plane. On the other hand, the P-model contains the entire H-plane when the whole reference circle is shown.