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In mathematics, a Laguerre plane is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane, named after the French mathematician Edmond Nicolas Laguerre.
Contents
- The classical real Laguerre plane
- The axioms of a Laguerre plane
- Finite Laguerre planes
- Miquelian Laguerre planes
- Ovoidal Laguerre planes
- References
Essentially the classical Laguerre plane is an incidence structure which describes the incidence behaviour of the curves
The classical real Laguerre plane
Originally the classical Laguerre plane was defined as the geometry of the oriented lines and circles in the real euclidean plane (see ). Here we prefer the parabola model of the classical Laguerre plane.
We define:
The incidence structure
The point set is
Points with the same x-coordinate cannot be connected by curves
Two points
For the description of the classical real Laguerre plane above two points
The incidence structure
Lemma:
Similar to the sphere model of the classical Moebius plane there is a cylinder model for the classical Laguerre plane:
The following mapping
The axioms of a Laguerre plane
The Lemma above gives rise to the following definition:
Let
Two points
Four points
From the definition of relation
Lemma: Relation
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
and call it the residue at point P.
In the plane model of the classical Laguerre plane
Theorem: Any residue of a Laguerre plane is an affine plane.
And the equivalent definition of a Laguerre plane:
Theorem: An incidence structure together with an equivalence relation
Finite Laguerre planes
The following incidence structure is a minimal model of a Laguerre plane:
Hence
For finite Laguerre planes, i.e.
Lemma: For any cycles
For a finite Laguerre plane
From combinatorics we get
Lemma: Let
Miquelian Laguerre planes
Unlike Moebius planes the formal generalization of the classical model of a Laguerre plane, i.e. replacing
Theorem: For a field
Similar to a Möbius plane the Laguerre version of the Theorem of Miquel holds:
Theorem of MIQUEL: For the Laguerre plane
(For a better overview in the figure there are circles drawn instead of parabolas)
The importance of the Theorem of Miquel shows the following theorem which is due to v. d. Waerden, Smid and Chen:
Theorem: Only a Laguerre plane
Because of the last Theorem
Remark: The minimal model of a Laguerre plane is miquelian.
Remark: A suitable stereographic projection shows:
Ovoidal Laguerre planes
There are a lot of Laguerre planes which are not miquelian (s. weblink below). The class which is most similar to miquelian Laguerre planes are the ovoidal Laguerre planes. An ovoidal Laguerre plane is the geometry of the plane sections of a cylinder which 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).