Showing papers in "Journal of Geometry in 1990"
••
TL;DR: The present paper completes a computer-free proof that every ovoid ofPG(3, 16) is an elliptic quadric.
Abstract: In a previous paper [8] the authors have shown that every ovoid ofPG(3, 16) is an elliptic quadric. The arguments used a computer and also depended on the computer-aided classification of hyperovals ofPG(2, 16) (see [3]). Recently (see [9]) the classification of hyperovals ofPG(2,16) has been obtained without the use of a computer. The present paper completes a computer-free proof that every ovoid ofPG(3,16) is an elliptic quadric.
50 citations
••
49 citations
••
TL;DR: In this article, strongly distributive multiplicative hyperrings are characterized and a non trivial example is provided giving a model for such hyperrings; also, a characterization theorem is proved.
Abstract: The aim of this paper is to characterize the strongly distributive multiplicative hyperrings. A non trivial example is provided giving a model for such hyperrings; also a characterization theorem is proved.
35 citations
••
TL;DR: In this article, it was shown that the only quadrics of finite type in Euclidean 3-space are the circular cylinder and the sphere, and that these quadrics are the only types of quadrics that can be represented by a circle.
Abstract: We show that the only quadrics of finite type in Euclidean 3-space are the circular cylinder and the sphere.
30 citations
••
TL;DR: In this paper, it was shown that the dual integral invariant of a closed ruled surface, the dual angle of pitch, corresponds to the dual spherical surface area described by the dual sphere indicatrix of the closed ruled surfaces.
Abstract: In this paper, it is shown that the dual integral invariant of a closed ruled surface, the dual angle of pitch, corresponds to the dual spherical surface area described by the dual spherical indicatrix of the closed ruled surface. So, new geometric interpretations of the real angle of pitch and the real pitch of a closed ruled surface, and some results are given.
25 citations
••
TL;DR: In this paper, it was shown that any affine-Cn-geometry with residues of grid type satisfying the intersection property is either an affine polar space or a standard quotient of an affined polar space.
Abstract: We prove that any affine-Cn-geometry with residues of grid type satisfying the Intersection Property is either an affine polar space or a standard quotient of an affine polar space in the sense of [7]. Dedicated to Professor Giuseppe Tallini on his 60th birthday
17 citations
••
TL;DR: Chebyshev and geodesic curvatures of the lines of an arbitrary net belonging to the n-dimensional space of Weyl Wn are introduced and the fundamental formulae in the case of an orthogonal coordinate net are obtained.
Abstract: Chebyshev and geodesic curvatures of the lines of an arbitrary net belonging to the n-dimensional space of Weyl Wn are introduced. Characteristics of the following special nets in Wn are found: strongly parallel, b-net, c-net and orthogonal (theorems 1.2, 1.3, 1.5, 1.6). Some properties of the Chebyshev nets, orthogonal b-nets and orthogonal c-nets, of the Chebyshev vectors of the second kind, of the orthogonal nets and nets containing Chebyshev subnets are established (theorems 1.1, 1.7, 1.8, 1.4). The fundamental formulae in the case of an orthogonal coordinate net are obtained. The spaces Wn containing one of the following special orthogonal nets — strongly parallel of the first kind, Chebyshev net of the second kind and b-net are defined.
17 citations
••
TL;DR: Using ideas from algebraic coding theory, a general notion of aderivation set for a projective plane is introduced and certain geometric codes are used to locate such sets.
Abstract: Using ideas from algebraic coding theory, a general notion of aderivation set for a projective plane is introduced. Certain geometric codes are used to locate such sets. These codes also lead to upper bounds for thep-ranks of incidence matrices of translation planes in terms of the dimensions of the associated codes.
15 citations
••
TL;DR: A class of ovals, called theOvali di Roma, is constructed in the non-Desarguesian finite Figueroa planes of odd order as discussed by the authors.
Abstract: A class of ovals, called theOvali di Roma, is constructed in the non-Desarguesian finite Figueroa planes of odd order.
10 citations
••
10 citations
••
TL;DR: In this paper, the authors present metrized versions of some of these properties and relationships and obtain new characterizations of real inner product spaces among complete, convex, externally convex metric spaces.
Abstract: Characterizations of real inner product spaces among normed linear spaces have been obtained by exploring properties of and relationships between various orthogonality relations which can be defined in such spaces. In the present paper the authors present metrized versions of some of these properties and relationships and obtain new characterizations of real inner product spaces among complete, convex, externally convex metric spaces.
••
••
TL;DR: In this paper, the theorem of ALEXANDROFF-LESTER was extended to non-regular metric vector spaces of characteristic 2, for which counterexamples and a new proof were presented.
Abstract: By presenting counterexamples and a new proof, we determine all metric vector spaces, for which the theorem of ALEXANDROFF-LESTER holds. In this context, the theorem will be extended to certain nonregular metric vector spaces of characteristic 2.
••
TL;DR: In this article, it was shown that every point in the plane which can be constructed by a compass and a ruler, given a setS of points, can also be constructed using a compass alone so that the following restriction is met.
Abstract: We show that every point in the plane which can be constructed by a compass and a ruler, given a setS of points, can be constructed using a compass alone so that the following restriction is met. LetO andK be two arbitrarily chosen distinct points ofS. Then every point is obtained as a proper intersection of two circles that are either completely symmetrical with respect to the lineOK or have both their centers on this line.
••
••
••
TL;DR: In this article, the radical R(T) of a planar ternary ring (PTR) T allows also to give an algebraic characterization of Junkers' multiple-valued halforderings of projective planes.
Abstract: The radical R(T) of a planar ternary ring (PTR) T, which plays an important role in the theory of halforderings and orderings of PTRs, allows also to give an algebraic characterization ofJunkers' multiple-valued halforderings of projective planes. As an application we obtain some information on the group of projectivities in projective planes.
••
TL;DR: In this article, the authors studied full orderings and half orderings in the class of linear ternary rings which arise from ordered planar near-fields by modification of multiplication and addition with methods given by F. R. Moulton and H. Naumann.
Abstract: There are studied full orderings and half-orderings in the wellknown class of linear ternary rings which arise from ordered planar near-fields by modification of multiplication and addition with methods given by F. R. Moulton [7] and H. Naumann [8] respectively.
••
TL;DR: In this article, the pole p of a Cayley-klein-plane has a unique hyperosculating logarthmic spiral, which consists of an affine and a metric part.
Abstract: At a pointk0 aC4-curve k of an affine Cayley-Klein-plane (CK-plane) has a unique hyperosculating logarthmic spiral We give a construction of the pole p of this spiral, which consists of an affine and a metric part This metric part is a similar one in the three CK — planes It is shown that this result is connected with results dealing with the center of the osculating circle given by RBereis in [2,p248]
••
TL;DR: In this article, the authors present a class of disconnected locally compact nearfields, all derived from tamely ramified extensions of local fields, which they call Dickson nearfields.
Abstract: It is the aim of this paper to present an extensive class of disconnected locally compact nearfields. They are all Dickson nearfields and derived from tamely ramified extensions of local fields.
••
TL;DR: In this article, the Euclidean classical problem of doubling the cube was investigated for the first time in hyperbolic geometry, and its partial solution was presented, along with some miscellaneous notes about straightedge and compass constructions.
Abstract: The Euclidean classical problem ofDuplicating the Cube is investigated for the first time inhyperbolic geometry, and its partial solution presented. The other two associated problems ofTrisecting an Angle andSquaring the Circle in the hyperbolic plane were solved classically, but a few additions to these solutions are presented, along with some miscellaneous notes about straightedge and compass constructions in the hyperbolic plane.
••
TL;DR: In this article, the bisectors of the edges of a triangle and the corresponding circumscribing circle are established as a special case of a theorem for triangles with weighted vertices where the edges are partitioned with circular arcs in the proportions of the weights.
Abstract: The theorem relating the bisectors of the edges of a triangle and the corresponding circumscribing circle is established as a special case of a theorem for triangles with weighted vertices where the edges are partitioned with circular arcs in the proportions of the weights. The circular arcs are established as being uniquely determined by the weights and the triangle, and are given by three circles with collinear centres. These circles either intersect in zero, one or two real points, these latter points being the triple points.
••
TL;DR: In this article, it was shown that n pencils of spheres which belong to the same bundle form a hexagonal-surface web, and it was concluded that any 4-pencils of the same sphere belonging to a bundle also form a polygonal surface web.
Abstract: In this work,it is shown that n pencils of spheres which belong to the same bundle form a hexagonal-surface-web.Firstly,4 pencils of spheres orthogonal to the same sphere are taken into consideration. Later,by means of a suitable transformation,the equations of these 4 pencils of spheres are written in their simplest form and the equation of the surface web is obtained.Then,it is concluded that any 4- pencils of spheres belonging to the same bundle form a hexagonal-surface-web.From this we conclude that a surface n-web which is formed by n pencils of spheres belonging to the same bundle is a hexagonal web.
••
TL;DR: In this article, the authors examined the connections between the isotropic invariants of a skew ruled surface and its kinematic image and obtained a correspondence between the theory of ruled surfaces and the elementary geometry of two pairs of plane curves with a common middle curve.
Abstract: TheClifford parallelism of the three-dimensional isotropic space J3(1) induces thekinematic mapping of an element of surface in J3(1) onto a pair of points in a fixed plane π0. By identifying a regular surface Φ with the manifold of its osculating elements of surface thekinematic image of Φ is defined and equivalent to an area preserving transformation of the plane π0. In this paper we will examine the connections between the isotropic invariants of a skew ruled surface Φ and its kinematic image. Since the striction line y of Φ exactly consists of the singular points of Φ the kinematic images of the osculating planes of y are considered in addition. In this way we obtain a correspondence between the theory of ruled surfaces and the elementary geometry of two pairs of plane curves with a common middle curve.
••
TL;DR: Some structure theorems on near rings are presented, the notion of a “coding set” of a near ring, which enables to construct (MDS)-codes, and the same problem for Laguerre codes is discussed.
Abstract: (MDS)- and Laguerre codes are closely related to geometry and can be used in order to construct certain finite incidence structures. Here we present some structure theorems on near rings, introduce the notion of a “coding set” of a near ring, which enables us to construct (MDS)-codes, and discuss the same problem for Laguerre codes. To find non trivial “Laguerre sets” in a near ring is much more difficult.
••
TL;DR: In this paper, it was shown that there exist projective planes of Lenz-Class III which are not isomorphic to any generalized Moulton plane, and a wide variety of non-Moulton planes of Class III.1 and III.2.
Abstract: As we have shown in [27] there do exist projective planes of Lenz-Class III which are not isomorphic to any (generalized) Moulton plane. We will go into some detail concerning the construction of these planes, present a wide variety of non-Moulton planes of Class III.1 and III.2, and determine their spaces of orderings. In particular, for any two-power z, we construct a Cartesian group C which satisfies Yaqub's criterion and whose distrubutor has index z in the multiplicative loop of C.
••
TL;DR: In this article, it was shown that a closed Jordan curve satisfying the infinitesimal rectangle property must be a convex curve of constant width, which generalizes the BesicovitchDanzer Theorem to closed Jordan curves.
Abstract: A.S. Besicovitch [1] and L. W. Danzer [3] characterized the circle among closed convex curves in the plane by the so-called rectangle property (for a precise definition see below). T. Zamfirescu [10] introduced an infinitesimal version of this property and proved, that a closed Jordan curve in the plane satisfying the infinitesimal rectangle property must be a convex curve of constant width. This generalizes the BesicovitchDanzer Theorem to closed Jordan curves.
••
TL;DR: In this article, the authors classify all finite linear spaces without three mutually parallel lines as a generalized projective plane (GPP) and show that such a space is a simple extension of a GP, or a complete inflated affine plane with a GP at infinity.
Abstract: We classify all finite linear spaces without three mutually parallel lines. Apart from two exceptions, such a space is necessarily a generalized projective plane, a simple extension of a generalized projective plane, or a complete inflated affine plane with a generalized projective plane at infinity.
••
TL;DR: The line sets of Baer subspaces have been studied in this article, where it is shown that any (t + 1)-dimensional subspace of a factor geometry P contains at least one line coplanar with at least another further line of the same subspace, and if a pointx of P is incident with at most two lines of a given line, then the points in the factor geometryP/x which are induced by the lines of x throughx form a blocking set of type (t, 1) in P/x.
Abstract: Generalizing a theorem of Beutelspacher and Seeger, we consider line sets\(\mathcal{L}\) inP=PG(2t + 1,q),t ∈ IN, with the following properties: (1) any (t + 1)-dimensional subspace ofP contains at least one line of\(\mathcal{L}\), (2) if a pointx ofP is incident with at least two lines of\(\mathcal{L}\) then the points in the factor geometryP/x which are induced by the lines of\(\mathcal{L}\) throughx form a blocking set of type (t, 1) inP/x, (3) any line of\(\mathcal{L}\) is coplanar with at least one further line of\(\mathcal{L}\). We will show that the examples of minimal cardinality are exactly the line sets of Baer subspaces ofP.