Laguerre Plane Research Articles

4-Dimensional point-transitive groups of automorphisms of 2-dimensional laguerre planes

It is shown that a 2-dimensional Laguerre plane that admits a closed connected 4-dimensional point-transitive group of automorphisms must be classical. Further, up to conjugacy in the automorphism group of the classical real Laguerre plane, all closed connected 4-dimensional point-transitive subgroups are determined.

Topological Antiregular Quadrangles

In a topological antiregular quadrangle whose point rows and line pencils are manifolds, the set of points collinear with three mutually noncollinear points depends continuously on the given points. This implies that the derivation of such a quadrangle yields a topological Laguerre plane.

Generalized quadrangles constructed from topological Laguerre planes

The Lie geometry of a finite-dimensional locally compact connected Laguerre plane is a topological generalized quadrangle.

The apollonius problem in flat laguerre planes

The number of circles of a flat Laguerre plane touching three given circles or points depends only on the given geometric configuration but not on the Laguerre plane.

The elation group of a 4-dimensional Laguerre plane

The elation group of a 4-dimensional Laguerre plane

Schliessungss�tze in Laguerre-Ebenen

For Laguerre planes Artzy [1] showed that a 4-point Pascal theorem on an oval leads to a configuration π, consisting of six points and five regular circles, which is equivalent Miquel's theorem. We represent some similar incidence assumptions which are again equivalent to π in each Laguerre plane. Besides, a uniform denotation to characterize different kinds of miquelian theorems in Benz planes is suggested.

On the structure of finite elation Laguerre planes

The miquelian Laguerre plane of order q (q being a prime power) is obtained as the geometry of non-trivial plane sections of a quadratic cone in the 3-dimensional projective space over GF(q). Similarly, an ovoidal Laguerre plane of order q is obtained as the geometry of non-trivial plane sections of a cone over an oval (not necessarily a conic) in the 3-dimensional projective space over GF(q).

On affine planes non-extensible to Laguerre planes and some related problems

Some examples of affine planes non-extensible to a Laguerre plane are studied and conditions for the uniqueness of a Laguerre extension are given.

Semiclassical topological flat Laguerre planes obtained by pasting along two parallel classes

A new rather large family of locally compact 2-dimensional topological Laguerre planes is introduced here. This family consists exactly of those Laguerre planes which can be obtained by pasting together two halves of the classical real Laguerre plane along two parallel classes suitably. Isomorphism classes and automorphisms of these planes are determined.

The octahedron theorem in Minkowski three-space a metric vector space proof of Miquel's theorem in the Laguerre plane

As is well known [1, p.480], the cycles and spears of the real Laguerre plane can be represented by the points and null planes of three-dimensional Minkowski space. Miquel's theorem in the Laguerre plane can then be expressed as an intrinsically interesting Minkowski space theorem the octahedron theorem. We outline the correspondence between the two theorems, and then give a metric vector space proof of the octahedron theorem, thereby providing an alternate proof of Miquel's theorem. We then discuss the generalization of both theorems to more general spaces.

Semiclassical Topological Flat Laguerre Planes Obtained by Pasting Along A Circle

A new rather large family of locally compact 2-dimensional topological Laguerre planes is introduced here. This family consists exactly of those Laguerre planes which can be obtained by pasting together two halves of the classical real Laguerre plane along a circle suitably. Isomorphism classes and automorphism groups of these planes are determined. Together with [9] this gives a complete classification of all semicalssical topological flat Lguerre planes.

Laguerre and Minkowski planes produced by dilatations

We show that each parabolic curve f in R2 produces a Laguerre plane $$\mathbb{L}(f)$$ if f and all its images under dilatations are cycles. Likewise, two hyperbolic curves f1,f2 produce a Minkowski planeM(f1,f2). We determine for which curves $$\mathbb{L}(f)$$ is miquelian resp. ovoidal, and for which pairs f1,f2,M(f1,f2) is miquelian resp. satisfies the rectangle axiom, thus providing many examples of non-embeddable planes.

A characterization of special laguerre planes and extended dual affine planes

A special Laguerre plane is a nondegenerate transversal 3-design such that the residue of each point is a dual affine plane. A special Laguerre plane is equivalent to an optimal code with three information digits and maximal length. An extended dual affine plane is an incidence structure (whose objects will be called points and blocks) such that the residue of each point is a dual affine plane, and each pair of points is in at least one block. Finite extended dual affine planes exist only of order 2, 4, and (dubiously) 10. We show that any finite incidence structure having the residue of each point a dual affine plane either is a transversal 3-design or has a block through each pair of points. Hence theorem: If a finite nondegenerate connected incidence structure θ has the residue of each point a dual affine plane, then θ is either an extended dual affine plane or a special Laguerre plane.

Zur Laguerreschen Ebenegeometrie

Zur Laguerreschen Ebenegeometrie

Semi Laguerre Planes

A Dembowski semi-plane is a semi-plane obtained from a projective plane by Dembowski's method [1]. A semi Laguerre plane is an incidence structure J = (P, B1 ∪ B2, I) for which: (a) every element of P is incident with one element of B1, (b) an element of B1 and an element of B2 are incident with at most one common element of P, (c) each residual space of J (with respect to B1) is a Dembowski semi-plane, (d) B2 ≠ ∅ and each element of B2 is incident with at least 4 elements of P. We prove that all semi Laguerre planes are substructures of Laguerre planes or special Laguerre planes (in the sense of Thas, Willems [3], [4]). Therefore, these incidence structures are related to optimal codes ([5], [6]).

Laguerre-Ebenen mit Symmetrien an Punktepaaren und Zykeln

Let ∞ be a point of a Laguerre plane, such that (S1) For any cycle containing ∞ there exists an automorphism of order 2 whose set of fixed points is exactly z. (S2) For any point X, not parallel to ∞, there exists an automorphism of order 2 whose set of fixed points is exactly {∞,X}.

A symmetry theorem for Laguerre planes

A 4-point Pascal theorem on an oval in a protective plane led to the symmetry theorem in the Minkowski plane, and this relation was used by the author in [2,3] to prove that the symmetry theorem is equivalent to Miquel's theorem and that it implies the tangency theorem (Beruhrsatz). The same special Pascal theorem now leads to an incidence theorem, π, for the Laguerre plane, which is again equivalent to Miquel's theorem. The configuration of π contains 6 points and 5 circles and is thus simpler than that of Miquel, although it does not have the intuitive symmetry properties of the corresponding configuration in the Minkowski plane.

A Characterization of Some Geometries of Chains

The geometries considered here are the Möbius plane M() (W. Benz [1]), the Laguerre plane L() (W. Benz and H. Mäurer [7]) and the Minkowski plane A() (W. Benz [5], G. Kaerlein [18]) over a field . All of them are geometries of an algebra with identity over a field. The characterization of the projective plane over a field by the proposition of Pappus first gave a close relation between algebraic and geometric structures. B. L. v. d. Waedern and L. J. Smid [28] presented a further example by characterizing the Möbius and Laguerre plane with incidence axioms and the "complete" proposition of Miquel.