Benz plane

Möbius plane

We assume that the real Euclidean plane and summarize the set of lines and the amount of circles to block a lot together. This design provides a very inhomogeneous incidence structure. Because by two points go exactly one line and any number of circles. The trick with which to embed this incidence structure in a homogeneous geometry, the following idea is: you add the set of points is another point that is set incident with each line. Now a block by exactly three points is uniquely determined. This " homogenized " geometry is called classical Möbius level (by August Ferdinand Möbius ). The still existing inhomogeneity of the description ( lines, circles ) can be eliminated by a spatial model. For by means of a stereographic projection, one can show that the classical Möbius plane to the geometry of the plane sections (circles) of a sphere ( in 3-dimensional space ) is isomorphic. Analogous to the axiomatic projective planes is called an incidence structure which is essentially the same incidence of behavior as the classical Möbius plane ( axiomatic ) Möbius plane. As expected, there are many Möbius planes that are different from the classical model.

Laguerre plane

Going back out of first and takes the curves of the form ( parabolas and straight lines ) as blocks, then the following homogenization proves useful: Take the curve to the new point, ie the point set consists of now. This geometry of the parabolas is called classical Laguerre plane ( after Edmond Laguerre ). ( It was originally formulated as the geometry of the directional lines and circles Both geometries are isomorphic to each other. . ) As in the classical Möbius level, there is also a spatial model: The classical Laguerre plane is the geometry of the plane sections on a vertical circular cylinder ( in ) isomorphic. An abstraction as the Möbius plane leads to the ( axiomatic ) Laguerre plane.

Minkowski plane

Assuming finally out of and increases the straights nor the hyperbolas as blocks are added, the following idea into a homogeneous incidence structure leads: you add any straight the point and at any hyperbola points added, ie the point set is made in this case. This geometry of the hyperbola is called the classical Minkowski plane ( after Hermann Minkowski ). As with the classical Möbius and Laguerre planes there is also a spatial model: The classical Minkowski plane is the geometry of the section cuts on a hyperboloid ( nondegenerate quadric of index 2 ) in 3-dimensional real projective space isomorphic. As with Möbius and Laguerre plane is reached by abstraction to the ( axiomatic ) Minkowski plane.

Level circuit geometries

Since the blocks in each of the three cases are projective circles ( nondegenerate conic sections ), one uses the collective term plane circular geometries. The word simply meant to indicate that there are also higher-dimensional Möbius, Laguerre and Minkowski geometries. In English literature, the planar circular geometries are also called Benz planes ( Benz planes) after the German mathematician Walter Benz.