# Laguerre plane

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.

Essentially the classical Laguerre plane is an incidence structure which describes the incidence behaviour of the curves $y=ax^{2}+bx+c$ , i.e. parabolas and lines, in the real affine plane. In order to simplify the structure, to any curve $y=ax^{2}+bx+c$ the point $(\infty ,a)$ is added. A further advantage of these completion is: The plane geometry of the completed parabolas/lines is isomorphic to the geometry of the plane sections of a cylinder (s. below).

## 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 $({\mathcal {P}},{\mathcal {Z}},\in )$ is called classical Laguerre plane.

Points with the same x-coordinate cannot be connected by curves $y=ax^{2}+bx+c$ . Hence we define:

For the description of the classical real Laguerre plane above two points $(a_{1},a_{2}),(b_{1},b_{2})$ are parallel if and only if $a_{1}=b_{1}$ . $\parallel$ is an equivalence relation, similar to the parallelity of lines.

The incidence structure $({\mathcal {P}},{\mathcal {Z}},\in )$ 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:

$\Phi :\ (x,z)\rightarrow ({\frac {x}{1+x^{2}}},{\frac {x^{2}}{1+x^{2}}},{\frac {z}{1+x^{2}}})=(u,v,w)\ .$ ## The axioms of a Laguerre plane

The Lemma above gives rise to the following definition:

B1: For any three points $A,B,C$ , pairwise not parallel, there is exactly one cycle $z$ which contains $A,B,C$ .
B2: For any point $P$ and any cycle $z$ there is exactly one point $P'\in z$ such that $P\parallel P'$ .
B3: For any cycle $z$ , any point $P\in z$ and any point $Q\notin z$ which is not parallel to $P$ there is exactly one cycle $z'$ through $P,Q$ with $z\cap z'=\{P\}$ ,
i.e. $z$ and $z'$ touch each other at $P$ .
B4: Any cycle contains at least three points, there is at least one cycle. There are at least four points not on a cycle.

From the definition of relation $\parallel$ and axiom B2 we get

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

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:

${\mathcal {A}}_{P}:=({\mathcal {P}}\setminus \{{\overline {P}}\},\{z\setminus \{{\overline {P}}\}\ |\ P\in z\in {\mathcal {Z}}\}\cup \{{\overline {Q}}\ |\ Q\in {\mathcal {P}}\setminus \{{\overline {P}}\},\in )$ and call it the residue at point P.

In the plane model of the classical Laguerre plane ${\mathcal {A}}_{\infty }$ is the real affine plane $\mathbb {R} ^{2}$ . In general we get

Theorem: Any residue of a Laguerre plane is an affine plane.

And the equivalent definition of a Laguerre plane:

## Finite Laguerre planes

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

${\mathcal {P}}:=\{A_{1},A_{2},B_{1},B_{2},C_{1},C_{2}\}\ ,$ ${\mathcal {Z}}:=\{\{A_{i},B_{j},C_{k}\}\ |\ i,j,k=1,2\}\ ,$ $A_{1}\parallel A_{2},\ B_{1}\parallel B_{2},\ C_{1}\parallel C_{2}\ .$ $|z_{1}|=|z_{2}|=|{\overline {P}}|+1$ .

From combinatorics we get

a) any residue ${\mathcal {A}}_{P}$ is an affine plane of order $n\quad ,$ b) $|{\mathcal {P}}|=n^{2}+n,$ c) $|{\mathcal {Z}}|=n^{3}.$ ## Miquelian Laguerre planes

Unlike Moebius planes the formal generalization of the classical model of a Laguerre plane, i.e. replacing $\mathbb {R}$ by an arbitrary field $K$ , leads in any case to an example of a Laguerre plane.

${\mathcal {P}}:=K^{2}\cup$ $(\{\infty \}\times K),\ \infty \notin K$ ,
${\mathcal {Z}}:=\{\{(x,y)\in K^{2}\ |\ y=ax^{2}+bx+c\}\cup \{(\infty ,a)\}\ |\ a,b,c\in K\}$ the incidence structure
${\mathcal {L}}(K):=({\mathcal {P}},{\mathcal {Z}},\in )$ is a Laguerre plane with the following parallel relation: $(a_{1},a_{2})\parallel (b_{1},b_{2})$ if and only if $a_{1}=b_{1}$ .

Similar to a Möbius plane the Laguerre version of the Theorem of Miquel holds:

Theorem of MIQUEL: For the Laguerre plane ${\mathcal {L}}(K)$ the following is true:

If for any 8 pairwise not parallel points $P_{1},\ldots ,P_{8}$ which can be assigned to the vertices of a cube such that the points in 5 faces correspond to concyclical quadruples than the sixth quadruple of points is concyclical, too.

(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 ${\mathcal {L}}(K)$ satisfies the theorem of Miquel.

Because of the last Theorem ${\mathcal {L}}(K)$ is called a miquelian Laguerre plane.

Remark: The minimal model of a Laguerre plane is miquelian.

It is isomorphic to the Laguerre plane ${\mathcal {L}}(K)$ with $K=GF(2)$ (field $\{0,1\}$ ).

Remark: A suitable stereographic projection shows: ${\mathcal {L}}(K)$ is isomorphic to the geometry of the plane sections on a quadric cylinder over field $K$ .

## 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 geometrty 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 no or 1 or two pints and 2) at any point there is a unique tangent. A simple oval in the real plane can be constructed by clueing together two suitable halves of different ellipses, such that the result is no conic. Even in the finite case there exist ovals (see quadratic set).