Conics in finite projective planes

There does not exist a general theory of conics in finite projective planes, because the many definitions of conics which are equivalent in desarguesian projective planes yield different types of conics in more general situations. Thus even the use of the word conic can lead to confusion, particularly in the finite case. This note is an attempt to clarify these various definitions and give as an example in a finite projective plane a von Staudt conic which is not an Ostrom conic. We conjecture that any finite projective plane admitting an Ostrom conic must be desarguesian.     

LDPC codes generated by conics in the classical projective plane

We construct various classes of low-density parity-check codes using point-line incidence structures in the classical projective plane PG(2,q). Each incidence structure is based on the various classes of points and lines created by the geometry of a conic in the plane. For each class, we prove various properties about dimension and minimum distance. Some arguments involve the geometry of two conics in the plane. As a result, we prove, under mild conditions, the existence of two conics, one entirely internal or external to the other. We conclude with some simulation data to exhibit the effectiveness of our codes.     

Conic blocking sets in Desarguesian projective planes

Define a conic blocking set to be a set of lines in a Desarguesian projective plane such that all conics meet these lines. Conic blocking sets can be used in determining if a collection of planes in projective three-space forms a flock of a quadratic cone. We discuss trivial conic blocking sets and conic blocking sets in planes of small order. We provide a construction for conic blocking sets in planes of non-prime order, and we make additional comments about the structure of these conic blocking sets in certain planes of even order.     

A remarkable configuration: A generalization of computergenerated results by S.-C. Chou and its connection with a plane figure by A. Emch

The paper's starting point are four theorems on conics which can be found in a collection of computer proved results by C.-S. Chou from 1987. It not only contains a generalization of two of Chou's results but also a plane figure consisting of points, lines and conics. A suitable notation will reveal a striking symmetry of this figure. Moreover, it turns out that a plane figure from 1940 found by A. Emch using algebraic methods is very similar to ours, which we obtained synthetically. As an application in finite geometry we have gone some way towards regarding our figure as a real projective model of the finite projective plane of order 4.Dedicated to Dr. J. F. Rigby on the occasion of his 65th birthday     

Zur Kennzeichnung von Quadriken unter den Kegelschnittflächen

A surface, generated by a one-parameter family of conics in projective 3-space, such that the tangent planes along a generating conic form a quadric cone, is called a surface of Blutel [1]. The surface is said to be of hyperbolic type, if the characteristic line of the plane of a generating conic intersects it in two different real points s₁, s₂. Formerly [5] it was shown that such a surface must be a quadric if it is unbranched along the curves, generated by s₁, s₂, these points not being stationary. In the present paper analogous results are established in the remaining cases when one or both points s₁, s₂ are fixed.

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Herrn Prof. Dr. K. Strubecker zum 80. Geburtstag gewidmet     

Shear planes

A shear plane is a 2n-dimensional stable plane admitting a quasi-perspective collineation group which is a vector group of the same dimension 2n and fixes no point. We show that all of these planes can be derived from a special kind of partial spreads by a construction analogous to the construction of (punctured) dual translation planes from compact spreads. Finally we give a criterion (and examples) for shear planes which are not isomorphic to an open subplane of a topological projective plane.     

Some Results on Contact Metric Manifolds

If the sectional curvatures of plane sections containing the characteristic vector field of a contact metric manifold M are non-vanishing, then we prove that a second order parallel tensor on M is a constant multiple of the associated metric tensor. Next, we prove for a contact metric manifold of dimension greater than 3 and whose Ricci operator commutes with the fundamental collineation that, if its Weyl conformal tensor is harmonic, then it is Einstein. We also prove that, if the Lie derivative of the fundamental collineation along the characteristic vector field on a contact metric 3-manifold M satisfies a cyclic condition, then M is either Sasakian or locally isometric to certain canonical Lie-groups with a left invariant metric. Next, we prove that if a three-dimensional Sasakian manifold admits a non-Killing projective vector field, it is of constant curvature 1. Finally, we prove that a conformally recurrent Sasakian manifold is locally isometric to a unit sphere.     

Die zweifachen Blutelschen Kegelschnittflächen

A surface in projective space generated by a one parameter family of conics is called a conic surface of Blutel if the tangent planes of taken along a generating conic, envelop a quadratic cone. If the conjugate curves (with respect to the generating conics) are conics, too, we call a two-fold Blutel's conic surface. In an earlier paper [4] it was shown that the planes of both conic families, the generating and the conjugate one, belong to a pencil, each. The present paper completes these investigations by integrating the derivative equations (3), (8), (9), (10). As a final result, a complete classification of all these surfaces is given. They are all algebraic of at most fourth order and furthermore—besides the quadrics and certain degenerate cases—they are complex projectively equivalent to the cyclides of Dupin.     

Characterization of projective planes of small prime orders

The method developed recently by the author to handle collineation groups of relative small order is used here to characterize the Desarguesian projective plane of order 11 by a collineation group isomorphic to the symmetric group of 4 letters. Also we prove that a finite projective plane of order 11, 13, or 17 is Desarguesian if it admits a collineation group, which is either strongly irreducible or non-abelian simple.     

On Angles and The Fundamental Theorems of Metric Geometry

In this paper we investigate the abstract angle measure for affine metric spaces. Common features and differences between orthogonal angles and angles with measure ≠ 0 are examined. It turns out that an affine collineation which maps angles with a certain fixed measure α ≠ 0,4 to angles with another fixed measure β is already a metric collineation in nearly all cases (fundamental theorem). An analogous result is stated for projective metric spaces. Some applications concerning minimal conditions for metric collineations are given.     

Visibility of the higher-dimensional central projection into the projective sphere

We can describe higher-dimensional classical spaces by analytical projective geometry, if we embed the d-dimensional real space onto a d + 1-dimensional real projective metric vector space. This method allows an approach to Euclidean, hyperbolic, spherical and other geometries uniformly [8]. To visualize d-dimensional solids, it is customary to make axonometric projection of them. In our opinion the central projection gives more information about these objects, and it contains the axonometric projection as well, if the central figure is an ideal point or an s-dimensional subspace at infinity. We suggest a general method which can project solids into any picture plane (space) from any central figure, complementary to the projection plane (space). Opposite to most of the other algorithms in the literature, our algorithm projects higher-dimensional solids directly into the two-dimensional picture plane (especially into the computer screen), it does not use the three-dimensional space for intermediate step. Our algorithm provides a general, so-called lexicographic visibility criterion in Definition and Theorem 3.4, so it determines an extended visibility of the d-dimensional solids by describing the edge framework of the two-dimensional surface in front of us. In addition we can move the central figure and the image plane of the projection, so we can simulate the moving position of the observer at fixed objects on the computer screen (see first our figures in reverse order). Supported by DAAD 2008 Multimedia Technology for Mathematics and Computer Science Education.     

Projective planes of order 12 do not have a collineation group of order 4

In this paper, we prove that there are no projective planes of order 12 admitting a collineation group of order 4. This yields that the order of any collineation group of a projective plane of order 12 is 1, 2, or 3.     

On Projective Planes of Order 15 Admitting a Collineation of Order 7

In this article we prove that there is no projective plane of order 15 admitting a collineation group of order 21. C. Y. Ho proved that there is no projective plane of order 15 admitting a collineation group of order 49. But his proof is incorrect. We also correct his error. The conclusion remains the same. We used a computer for our research.     

On the Fundamental Groups of Galois Covering Spaces of the Projective Plane

We compute fundamental groups of the complements of a class of real curves in the complex projective plane. As a result, we obtain a new Zariski pair for arrangements of conics. As an application, we give a method for the computations of the fundamental groups of resolutions of Galois covering spaces of the projective plane ramifying along a special type of curves.     

Collineation groups of the intersection of two classical unitals

Kestenband proved in [12] that there are only seven pairwise non‐isomorphic Hermitian intersections in the desarguesian projective plane PG(2, q) of square order q. His classification is based on the study of the minimal polynomials of the matrices associated with the curves and leads to results of purely combinatorial nature: in fact, two Hermitian intersections from the same class might not be projectively equivalent in PG(2, q) and might have different collineation groups. The projective classification of Hermitian intersections in PG(2, q) is the main goal in this paper. It turns out that each of Kestenband's classes consists of projectively equivalent Hermitian intersections. A complete classification of the linear collineation groups preserving a Hermitian intersection is also given. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 445–459, 2001     

A classification and construction of entirely circular cubics in the hyperbolic plane

If each intersection point of a third order curve with the absolute conic of the hyperbolic plane is a tangential point, this curve will be called an entirely circular cubic. According to this definition a rough classification of such curves is given into four main types and nine sub-types. Some of them are constructed by a (1,2) or (1,1) mapping and the others are constructed by the generalized quadratic hyperbolic inversion. Thus we extend and complete Palman's paper [5] in a synthetic way. This revised version was published online in June 2006 with corrections to the Cover Date.     

Extending Homomorphisms of Dense Projective Subplanes by Continuity

Let Ψ be a dense projective subplane of a topological projective plane Π. We show that a continuous homomorphism a of Ψ is extendable to a continuous homomorphism of Π if and only if there is a line Z of Ψ such that the restriction of α to the Ψ-points of Z is continuously extendable to some mapping defined on all Π-points of Z. In particular, each projective collineation of Ψ is extendable to a projective collineation of Π yielding the well-known result that (z, A)-transitivity of Ψ extends to (z, A)-transitivity of Π.     

Zentrale und axiale Kollineationen in projektiven Hjelmslev-Ebenen

Zusammenfassung Es gibt in projektiven Hjelmslev-Ebenen zentrale (axiale) Kollineationen, die keine Achse (Zentrum) haben. Das Produkt zweier zentraler Kollineationen s und t mit gemeinsamer Achse kann eine axiale Kollineation ohne Zentrum sein oder auch eine axiale Kollineation mit einem Zentrum, daß auf keiner Verbindungsgeraden der Zentren von s und t liegt.

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In ordinary projective planes every central collineation has an axis and every collineation with an axis is central. We prove in this paper, that this proposition doesn't hold in projective Hjelmslev-planes. We construct a projective Hjelmslev-plane and collineations with centers Pand Q on a common axis g such that the product of these collineations has no center but the axis g. In the dual plane we get a central collineation without an axis.

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Herrn R. Artzy zum siebzigsten Geburtstag gewidmet