In mathematics, the classical Möbius plane (named after August Ferdinand Möbius) is the Euclidean plane supplemented by a single point at infinity. It is also called the inversive plane because it is closed under inversion with respect to any generalized circle, and thus a natural setting for planar inversive geometry. An inversion of the Möbius plane with respect to any circle is an involution which fixes the points on the circle and exchanges the points in the interior and exterior, the center of the circle exchanged with the point at infinity. In inversive geometry a straight line is considered to be a generalized circle containing the point at infinity; inversion of the plane with respect to a line is a Euclidean reflection. More generally, a Möbius plane is an incidence structure with the same incidence relationships as the classical Möbius plane. It is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane. Affine planes are systems of points and lines that satisfy, amongst others, the property that two points determine exactly one line. This concept can be generalized to systems of points and circles, with each circle being determined by three non-collinear points. However, three collinear points determine a line, not a circle. This drawback can be removed by adding a point at infinity to every line. If we call both circles and such completed lines cycles, we get an incidence structure in which every three points determine exactly one cycle. In an affine plane the parallel relation between lines is essential. In the geometry of cycles, this relation is generalized to the touching relation. Two cycles touch each other if they have just one point in common. This is true for two tangent circles or a line that is tangent to a circle. Two completed lines touch if they have only the point at infinity in common, so they are parallel. The touching relation has the property for any cycle , point on and any point not on there is exactly one cycle containing points and touching (at point ).

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Laguerre plane
In mathematics, a Laguerre plane is one of the three types of Benz plane, which are the Möbius plane, Laguerre plane and Minkowski plane. Laguerre planes are named after the French mathematician Edmond Nicolas Laguerre. The classical Laguerre plane is an incidence structure that describes the incidence behaviour of the curves , i.e. parabolas and lines, in the real affine plane. In order to simplify the structure, to any curve the point is added.
Minkowski plane
In mathematics, a Minkowski plane (named after Hermann Minkowski) is one of the Benz planes (the others being Möbius plane and Laguerre plane). Applying the pseudo-euclidean distance on two points (instead of the euclidean distance) we get the geometry of hyperbolas, because a pseudo-euclidean circle is a hyperbola with midpoint . By a transformation of coordinates , , the pseudo-euclidean distance can be rewritten as . The hyperbolas then have asymptotes parallel to the non-primed coordinate axes.
Finite geometry
A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity.
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