In mathematics, a Laguerre plane is one of the Benz planes: the Möbius plane, Laguerre plane and Minkowski plane, named after the French mathematician Edmond Nicolas Laguerre. Essentially the classical Laguerre plane is an incidence structure that describes the incidence behaviour of the curves , i.e. parabolas and lines, in the realaffine plane. In order to simplify the structure, to any curve the point is added. A further advantage of this completion is: The plane geometry of the completed parabolas/lines is isomorphic to the geometry of the plane sections of a cylinder.
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. Here we prefer the parabola model of the classical Laguerre plane. We define: the set of points, the set of cycles. The incidence structure is called classical Laguerre plane. The point set is plus a copy of . Any parabola/line gets the additional point . Points with the same x-coordinate cannot be connected by curves. Hence we define: Two points are parallel if or there is no cycle containing and. For the description of the classical real Laguerre plane above two points are parallel if and only if. is an equivalence relation, similar to the parallelity of lines. The incidence structure 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 . The following mapping is a projection with center that maps the x-z-plane onto the cylinder with the equation, axis and radius
The points appear not as images.
projects the parabola/line with equation into the plane. So, the image of the parabola/line is the plane section of the cylinder with a non perpendicular plane and hence a circle/ellipse without point. The parabolas/line are mapped onto circles.
A line is mapped onto a circle/Ellipse through center and a parabola onto a circle/ellipse that do not contain.
The axioms of a Laguerre plane
The Lemma above gives rise to the following definition: Let be an incidence structure with point set and set of cycles.
Two points are parallel if or there is no cycle containing and.
is called Laguerre plane if the following axioms hold: Four points are concyclic if there is a cycle with. From the definition of relation and axiom B2 we get Lemma: Relation is an equivalence relation. Following the cylinder model of the classical Laguerre-plane we introduce the denotation: a) For we set. b) An equivalence class is called generator. For the classical Laguerre plane a generator is a line parallel to the y-axis or a line on the cylinder. 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 is the real affine plane . In general we get 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 on is a Laguerre plane if and only if for any point the residue is an affine plane.
Finite Laguerre planes
The following incidence structure is a minimal model of a Laguerre plane: Hence and For finite Laguerre planes, i.e., we get: Lemma: For any cycles and any generator of a finite Laguerre plane we have: For a finite Laguerre plane and a cycle the integer is called order of. From combinatorics we get Lemma: Let be a Laguerre—plane of order. Then
Miquelian Laguerre planes
Unlike Moebius planes the formal generalization of the classical model of a Laguerre plane, i.e. replacing by an arbitrary field, leads in any case to an example of a Laguerre plane. Theorem: For a field and Similar to a Möbius plane the Laguerre version of the Theorem of Miquel holds: Theorem of MIQUEL: For the Laguerre plane the following is true: 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 satisfies the theorem of Miquel. Because of the last Theorem is called a miquelian Laguerre plane. Remark: The minimal model of a Laguerre plane is miquelian.
Remark: A suitable stereographic projection shows: is isomorphic to the geometry of the plane sections on a quadric cylinder over field.
Ovoidal Laguerre planes
There are a lot of Laguerre planes that are not miquelian 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.
