Laguerre plane

In mathematics, a Laguerre plane is one of the three types of Benz plane, which are the Möbius plane, Laguerre plane and Minkowski plane. Laguerre planes are named after the French mathematician Edmond Nicolas Laguerre.

The classical Laguerre plane is an incidence structure that 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 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).

The points $(0,1,a)$ (line on the cylinder through the center) appear not as images.

$\Phi$ projects the parabola/line with equation $z=ax^{2}+bx+c$ into the plane $w-a=bu+(a-c)(v-1)$ . 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 $(0,1,a)$ . The parabolas/line $z=ax^{2}+a$ are mapped onto (horizontal) circles.

A line(a=0) is mapped onto a circle/Ellipse through center $(0,1,0)$ and a parabola ( $a\neq 0$ ) onto a circle/ellipse that do not contain $(0,1,0)$ .

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:

${\mathcal {P}}:=\mathbb {R} ^{2}\cup (\{\infty \}\times \mathbb {R} ),\ \infty \notin \mathbb {R} ,$ the set of points, ${\mathcal {Z}}:=\{\{(x,y)\in \mathbb {R} ^{2}\mid y=ax^{2}+bx+c\}\cup \{(\infty ,a)\}\mid a,b,c\in \mathbb {R} \}$ the set of cycles.

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

The point set is $\mathbb {R} ^{2}$ plus a copy of $\mathbb {R}$ (see figure). Any parabola/line $y=ax^{2}+bx+c$ gets the additional point $(\infty ,a)$ .

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

Two points $A,B$ are parallel ( $A\parallel B$ ) if $A=B$ or there is no cycle containing $A$ and $B$ .

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:

$({\mathcal {P}},{\mathcal {Z}},\in )$ is isomorphic to the geometry of plane sections of a circular cylinder in $\mathbb {R} ^{3}$ .

The following mapping $\Phi$ is a projection with center $(0,1,0)$ that maps the x-z-plane onto the cylinder with the equation $u^{2}+v^{2}-v=0$ , axis $(0,{\tfrac {1}{2}},..)$ and radius $r={\tfrac {1}{2}}\ :$

Ovoidal Laguerre planes[edit]

There are many Laguerre planes that are not miquelian (see weblink below). The class that is most similar to miquelian Laguerre planes is the ovoidal Laguerre planes. 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).

Laguerre transformations

in the Encyclopedia of Mathematics

Laguerre plane

Ovoidal Laguerre planes[edit]

Laguerre transformations

Benz plane

Lecture Note Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes, pp. 67