In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Laguerre plane is one of the three types of
Benz plane
In mathematics, a Benz plane is a type of 2-dimensional geometrical structure, named after the German mathematician Walter Benz. The term was applied to a group of objects that arise from a common axiomatization of certain structures and split int ...
, which are the
Möbius plane In mathematics, a Möbius plane (named after August Ferdinand Möbius) is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane. The classical example is based on the geometry of lines and circles in the real affine plane.
A s ...
, Laguerre plane and
Minkowski plane
In mathematics, a Minkowski plane (named after Hermann Minkowski) is one of the Benz planes (the others being Möbius plane and Laguerre plane).
Classical real Minkowski plane
Applying the pseudo-euclidean distance d(P_1,P_2) = (x'_1-x'_2) ...
. Laguerre planes are named after the
French mathematician
Edmond Nicolas Laguerre.
The classical Laguerre plane is an
incidence structure
In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore al ...
that describes the incidence behaviour of the curves
, i.e. parabolas and lines, in the
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
affine plane
In geometry, an affine plane is a two-dimensional affine space.
Examples
Typical examples of affine planes are
* Euclidean planes, which are affine planes over the reals equipped with a metric, the Euclidean distance. In other words, an affine pl ...
. In order to simplify the structure, to any curve
the point
is added. A further advantage of this completion is that the plane geometry of the completed parabolas/lines is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the geometry of the
plane section
In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections. The ...
s of a
cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...
(see 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 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
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
, similar to the parallelity of lines.
The incidence structure
has the following properties:
Lemma:
:* For any three points
, pairwise not parallel, there is exactly one cycle
containing
.
:* For any point
and any cycle
there is exactly one point
such that
.
:* For any cycle
, any point
and any point
that is not parallel to
there is exactly one cycle
through
with
, i.e.
and
touch each other at
''.
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) 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 (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:
:B1: For any three points
, pairwise not parallel, there is exactly one cycle
that contains
.
:B2: For any point
and any cycle
there is exactly one point
such that
.
:B3: For any cycle
, any point
and any point
that 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
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each 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
In geometry, an affine plane is a two-dimensional affine space.
Examples
Typical examples of affine planes are
* Euclidean planes, which are affine planes over the reals equipped with a metric, the Euclidean distance. In other words, an affine pl ...
.
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
:a) any residue
is an affine plane of order
b)
c)
Miquelian Laguerre planes
Unlike Moebius planes the formal generalization of the classical model of a Laguerre plane, i.e. replacing
by an arbitrary field
, always leads to an example of a Laguerre plane.
Theorem:
For a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
and
:
,
:
the incidence structure
:
is a Laguerre plane with the following parallel relation:
if and only if
.
Similarly 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
that can be assigned to the vertices of a cube such that the points in 5 faces correspond to concyclical quadruples then 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 in 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".
The minimal model of a Laguerre plane is miquelian. It is isomorphic to the Laguerre plane
with
(field
).
A suitable
stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
shows that
is isomorphic to the geometry of the plane sections on a quadric cylinder over field
.
Ovoidal Laguerre planes
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
An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one or ...
instead of a non degenerate conic. An oval is a
quadratic set In mathematics, a quadratic set is a set of points in a projective space that bears the same essential incidence properties as a quadric (conic section in a projective plane, sphere or cone or hyperboloid in a projective space).
Definition of a qu ...
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 In mathematics, a quadratic set is a set of points in a projective space that bears the same essential incidence properties as a quadric (conic section in a projective plane, sphere or cone or hyperboloid in a projective space).
Definition of a qu ...
).
See also
*
Laguerre transformations
Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigat ...
References
External links
Benz planein the ''
Encyclopedia of Mathematics
The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics.
Overview
The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduat ...
''
Lecture Note ''Planar Circle Geometries'', an Introduction to Moebius-, Laguerre- and Minkowski Planes, pp. 67
Planes (geometry)