# 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 , i.e. parabolas and lines, in the real affine plane. In order to simplify the structure, to any curve the point 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 ^{[1]}). 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 (see figure). 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 (line on the cylinder through the center) apper 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 (horizontal) circles. - A line(a=0) 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:

**B3:**For any cycle , any point and any point which is not parallel to there is exactly one cycle through with ,- i.e. and
**touch**each other at .

**B4:**Any cycle contains at least three points, there is at least one cycle. There are at least four points not on a cycle.

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 (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 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:

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

- ,
- the incidence structure
- is a Laguerre plane with the following parallel relation: if and only if .

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:

- If for any 8 pairwise not parallel points 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 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 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).