The models of hyperbolic geometry

Let’s see some basics: the hyperbolic models are for speculative purposes, because they don’t represent the hyperbolic plaane, because this(the hyperbolic plane) exists in the curved universe. We’ll talk about two models:the Poincaré-disk, and the upper half-plane, let’s see how they work!. On the Poincaré-disk, lines are circles, that are perpendicular to the basic circle, while on the upper half-plane, whose centre is on the lower terminal line.

Now let’s take a look at several geometric components. Let’s start with the angles. In hyperbolic geometry, we consider angles as: enclosed by two half-lines with mutual zero point (like Euclidean geometry). But in hyperbolic geometry, lines are periods of circles’. That’s why the two angle-shafts are lines starting from the zero point, and thus the two angle shafts are straight lines drawn from the starting point which are tangential to the two segments. Now, when we know what an angle is like, let’s look at the best-known theorem of hyperbolic geometry, which declares that the sum of the angles of a triangle is always less than 180°. This –as the picture shows- comes from the definition of angles.

Now, when we know what’s like an angle, let’s look after the hyperbolic geometry’s best-known theorem, which declares, that triangles’ inside-angles summary is always less than 180o. This –as the picture shows- come’s from the definitoin of angles.

Now, there are some theorems that we’ve learned in our Maths lessons. These theorems are true in Euclidean geometry and in hyperbolic geometry, too.

Common triangles:

Triangle heights intersect in one point.
Triangle medians intersect in one point.
Trianglebisectrices meet in one point.
In any kind of triangle, two sides are longer than the third side.
This next theorem isn’t true in hyperbolic geometry, because we calculate area in some other way in hyperbolic geometry.
In ABC triangle, AB*mc=AC*mb=BC*ma.

Now let’s see Euclid’s paralell axiom! This is true in the Euclidean plane: if there’s a p point and an e line, there’s one, and only one line going through the p point and it never crosses e. This isn’t true in the hyperbolic plane (we consider two lines parallel, if they are in one plane, and they never cross each other). The axiom sounds like this in the hyperbolic geometry: if there’s a line and a point, there are at least two lines that go through the point and never cross the line. Because on the hyperbolic plane the hyperbolic paralell axiom conforms with the the other four axioms of Euclidean geometry, János Bolyai wrote this to his father: ”… I made a new world out of nothing.”

Now, that we understood the main differences between Euclidean and hyperbolic geometry, we’ll take a look at a bit more complicated topic: the Poincaré-disk, and the upper half plane!
We consider a model a model of hyperbolic geometry if the relations relative to points, angles etc.are true on that model.

The Poincaré-disk is practically a C circle, that exists on the Euclidean plane.

Its points are the D-points: Euclidean points of the interior of C
Its lines are the D-lines- A D-line is either (1) the intersection of Ω and C⊥, or (2) the intersection of Ω and a diameter of C
Its circles are the D-circles- a D-circle is defined as the set of all D-points that are equal D-distance from a given D-point.

The upper Half-plane is similar. Its base is the ST line (which is the x axis).

Its points are the H-points: H-points are Euclidean points on one side of line ST. Let Ψ denote the set of all H-points.
Its lines are the H-lines: An H-line is either (1) a semicircle within Ψ, and with center on ST, or (2) the intersection of Ψ and a perpendicular to ST.