Laguerre plane

, i.e. parabolas and lines, in the real affine plane.

In order to simplify the structure, to any curve

A further advantage of this completion is that the plane geometry of the completed parabolas/lines is isomorphic to the geometry of the plane sections of a cylinder (see below).

Originally the classical Laguerre plane was defined as the geometry of the oriented lines and circles in the real Euclidean plane (see [1]).

Here we prefer the parabola model of the classical Laguerre plane.

is called classical Laguerre plane.

Points with the same x-coordinate cannot be connected by curves

For the description of the classical real Laguerre plane above two points

is an equivalence relation, similar to the parallelity of lines.

has the following properties: Lemma: Similar to the sphere model of the classical Moebius plane there is a cylinder model for the classical Laguerre plane:

is isomorphic to the geometry of plane sections of a circular cylinder in

that maps the x-z-plane onto the cylinder with the equation

be an incidence structure with point set

is called Laguerre plane if the following axioms hold: Four points

Following the cylinder model of the classical Laguerre-plane we introduce the denotation: a) For

For the classical Laguerre plane a generator is a line parallel to the y-axis (plane model) or a line on the cylinder (space model).

The connection to linear geometry is given by the following definition: For a Laguerre plane

we define the local structure and call it the residue at point P. In the plane model of the classical Laguerre plane

The following incidence structure is a "minimal model" of a Laguerre plane: Hence

Then Unlike Moebius planes the formal generalization of the classical model of a Laguerre plane, i.e. replacing

and Similarly to a Möbius plane the Laguerre version of the Theorem of Miquel holds: Theorem of Miquel: For the Laguerre plane

the following is true: (For a better overview in the figure there are circles drawn instead of parabolas) The importance of the Theorem of Miquel shows in the following theorem, which is due to v. d. Waerden, Smid and Chen: Theorem: Only a Laguerre plane

is called a "Miquelian Laguerre plane".

The minimal model of a Laguerre plane is miquelian.

A suitable stereographic projection shows that

is isomorphic to the geometry of the plane sections on a quadric cylinder over field

An ovoidal Laguerre plane is the geometry of the plane sections of a cylinder that is constructed by using an oval instead of a non degenerate conic.

An oval is a quadratic set and bears the same geometric properties as a non degenerate conic in a projective plane: 1) a line intersects an oval in zero, one, or two points and 2) at any point there is a unique tangent.

A simple oval in the real plane can be constructed by glueing together two suitable halves of different ellipses, such that the result is not a conic.

Even in the finite case there exist ovals (see quadratic set).