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.[br][br]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.[br]

[br][table] [tr] [td] [b]The modeled concept: [/b][br][/td] [td] [b]The model (in the Euclidean plane): [/b][br][/td] [/tr] [tr] [td] [color=#9900ff][i]H-plane[/i][/color][br][/td] [td] [color=#1e84cc] [/color][color=#0000ff]Disk[/color][br][/td] [/tr] [tr] [td][i][color=#9900ff]H-point [/color][/i][br][/td] [td] I[color=#0000ff]nner point of the disk[/color][br][/td] [/tr] [tr] [td][i][color=#9900ff]H-line [/color][/i][br][/td] [td] [color=#0000ff]An arc of a circle which is perpendicular to the boundary of the disk,[br] the arc is inside the disk[/color][br][/td] [/tr] [tr] [td][i][color=#9900ff]H-segment[/color][/i][br][/td] [td][color=#0000ff]An arc between two points of an H-line[/color][br][/td] [/tr] [tr] [td][i][color=#9900ff]H-reflection[/color][/i][br][/td] [td][color=#0000ff]An inversion with respect to an H-line or H-segment[br][/color][/td] [/tr][/table]

This model of the hyperbolic geometry will be called [i]P-model [/i]after its creator [url=https://en.wikipedia.org/wiki/Henri_Poincar%C3%A9]Henri Poincaré[/url] (1854-1912). [br]By using dynamic geometry the entire H-line can be displayed, including the border of the disk. [br][br]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.

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.[br][br][br]