Hyperbolic geometry

The hyperbolic geometry as an example of a non-Euclidean geometry is obtained when one of the axioms of absolute geometry instead of the parallel axiom that characterizes the Euclidean geometries that this contradictory " hyperbolic axiom ", it is. The hyperbolic axiom states that there is a line g and a point P (which is not on g) not as in Euclidean geometry only just one but at least two lines are (h and i) that pass through P and g are parallel. The fact that two lines are "parallel" to each other, but here means only that they lie in the same plane and have no points in common, not that they all have the same distance ( h and i have only one common point P).

It can be shown that there is then an arbitrary straight line g through any point not on l infinitely many non-cutting end is ( "Parallel " ), which lie in the straight line through the point and some level. Two of them are in a boundary layer and hot cross parallel (also: horoparallel ) to the line, while the remaining lines on parallel (also: hyper parallel) are called.

Representations of the real hyperbolic plane

There are various ways in which the real hyperbolic plane can be represented in the real Euclidean plane. Most of them can be generalized to higher dimensions.

On each of these types the same abstract hyperbolic geometry is represented: the real hyperbolic plane. It is therefore possible to convert between these representations and statements in purely hyperbolic geometry are used by the "model" independent. Usually this is called in mathematics of different models, if two non- isomorphic structures satisfy the same system of axioms. In this respect, describe the following " models " have the same structure, that are just different representations of a model. These representations are, however, always referred to in the literature as models, as well as here. For hyperbolic planes over other bodies and more than two-dimensional hyperbolic spaces, see Metric absolute geometry.

Circular disk model of Beltrami and Klein

In this developed by Eugenio Beltrami and Felix Klein representation is:

• The hyperbolic plane is modeled by an open circular disk.
• Hyperbolic lines are modeled by tendons.
• Length is defined by a particular distance function ( the angles are different from the Euclidean values).

This representation is also known under the name " coasters geometry ".

Distance function

If A and B are two points of the circular disk, the line passing through A and B tendon meets the circle in two points R and S. The hyperbolic distance of A and B is now defined using the double ratio:

Poincarésches circular disk model

When the going back to Beltrami disc model of Henri Poincaré applies:

• The hyperbolic plane is an open disk (usually the unit circle ) models.
• Hyperbolic lines are modeled by arcs ( and diameter), which are perpendicular to the edge.
• The hyperbolic angle measurements corresponding to the Euclidean angle measurement, whereby the angle between two circular arcs is determined by the tangents to the point of intersection.
• The hyperbolic length is measured by a specific distance function.

Distance function

Let A and B be two points of the circular disc. Summarizing the plane as a complex number plane, the corresponding points A, B are complex numbers a, b ​​. The hyperbolic spacing of A and B will now be defined by means of these complex numbers:

Poincarésches half-plane model

When the going back to Beltrami half-plane model of Henri Poincaré applies:

• The hyperbolic plane is modeled by the upper half- plane (y > 0).
• Hyperbolic lines are modeled by circular arcs ( and half-line ), which are perpendicular to the x-axis.
• The hyperbolic angle measurements corresponding to the Euclidean angle measurement, whereby the angle between two circular arcs is determined by the tangents to the point of intersection.

Distance function

The distance between two points of the upper half-plane is calculated by the following formula:

Hyperboloid model

Dating back to Poincaré Hyperboloidmodell embeds the hyperbolic plane in the three-dimensional Minkowski space.

Erlanger program

For the purposes of Felix Klein's Erlanger program hyperbolic geometry is the geometry of

The Beltrami -Klein model shows that hyperbolic geometry can be regarded as part of projective geometry.

Triangle

In the real hyperbolic geometry the sum of angles in a triangle is always less than π ( 180 degrees;, or two rights if you want to avoid is the angle ). For very large triangles it can be arbitrarily small. The area of ​​the triangle is calculated by Johann Heinrich Lambert's formula:

Wherein α, β and γ, the respective angle Δ the surface and the constant C is a scaling factor. The scaling factor C depends on the unit system used to set and basically equal to 1. If the factor C is negative, it is called a (positive) Gaussian curvature. Similarly, Thomas Harriot defined previously in the year 1603, the formula

For the area of ​​a triangle on a spherical surface that is formed of circles with the same radius as the ball. Hiebei applies the relationship

As a positive value for C is required for hyperbolic geometry, it must be due to the case of R

Be an imaginary radius.