Laguerre plane
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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 y=ax^2+bx+c , 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 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 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: \mathcal P:=\R^2\cup (\\times\R), \ \infty \notin \R, the set of points, \mathcal Z:=\ the set of cycles. The incidence structure (\mathcal P,\mathcal Z, \in) is called classical Laguerre plane. The point set is \R^2 plus a copy of \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 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 (\mathcal P,\mathcal Z, \in) has the following properties: Lemma: :* For any three points A,B,C, pairwise not parallel, there is exactly one cycle z containing A,B,C. :* For any point P and any cycle z there is exactly one point P'\in z such that P\parallel P'. :* For any cycle z, any point P\in z and any point Q\notin z that is not parallel to P there is exactly one cycle z' through P,Q with z\cap z'=\, i.e. z and z' touch each other at P''. 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 \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,..) and radius r=\tfrac\ : :\Phi: \ (x,z) \rightarrow (\frac,\frac,\frac)=(u,v,w)\ . *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\ne0) onto a circle/ellipse that do not contain (0,1,0).


The axioms of a Laguerre plane

The Lemma above gives rise to the following definition: Let \mathcal L:=(\mathcal P,\mathcal Z, \in) be an incidence structure with point set \mathcal P and set of cycles \mathcal Z.
Two points A,B are parallel (A\parallel B) if A=B or there is no cycle containing A and B.
\mathcal L is called Laguerre plane if the following axioms hold: :B1: For any three points A,B,C, pairwise not parallel, there is exactly one cycle z that 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 that is not parallel to P there is exactly one cycle z' through P,Q with z\cap z'=\, : 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. Four points A,B,C,D are concyclic if there is a cycle z with A,B,C,D \in z. From the definition of relation \parallel and axiom B2 we get Lemma: Relation \parallel 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 P\in \mathcal P we set \overline:=\. b) An equivalence class \overline 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 \mathcal L:=(\mathcal P,\mathcal Z, \in) we define the local structure : \mathcal A _P:= (\mathcal P\setminus\,\ \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 \R^2. 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 \parallel on \mathcal P is a Laguerre plane if and only if for any point P the residue \mathcal A _P is an affine 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 \ . Hence , \mathcal P, = 6 and , \mathcal Z, =8 \ . For finite Laguerre planes, i.e. , \mathcal P, <\infty, we get: Lemma: For any cycles z_1,z_2 and any generator \overline{P} of a ''finite'' Laguerre plane \mathcal L:=(\mathcal P,\mathcal Z, \in) we have: : , z_1, =, z_2, =, \overline{P}, +1. For a finite Laguerre plane \mathcal L:=(\mathcal P,\mathcal Z, \in) and a cycle z\in \mathcal Z the integer n:=, z, -1 is called order of \mathcal L. From combinatorics we get Lemma: Let \mathcal L:=(\mathcal P,\mathcal Z, \in) be a Laguerre—plane of order n. Then :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 \R by an arbitrary field K, always leads to an example of a Laguerre plane. Theorem: For a
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K and : \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. Similarly 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 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 \mathcal L(K) satisfies the theorem of Miquel. Because of the last theorem \mathcal L(K) is called a "Miquelian Laguerre plane". 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\}). 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 \mathcal L(K) is isomorphic to the geometry of the plane sections on a quadric cylinder over field K .


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 plane
in 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)