Showing papers in "Journal of Geometry in 1984"
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TL;DR: In this article, the consequences of the congruence axioms for other incidence structures e.g. finite or half ordered ones, and discuss the question when the 3-reflection theorem is valid.
Abstract: Euclidean planes are characterized as affine planes with a congruence relation on the set of the pairs of points. Furthermore we study the consequences of the congruence axioms for other incidence structures e.g. finite or half ordered ones, and discuss the question when the 3-reflection theorem is valid.
24 citations
17 citations
TL;DR: In this article, a criterion for injective lineations to be full and surjective is given; in connection with this, some bad injective lines are constructed and studied.
Abstract: Some geometric properties of full lineations are given and used to extend an affine full lineation to the projective envelope of its domain. A criterion for injective lineations to be full and surjective is given; in connection with this some bad injective lineations are constructed and studied.
17 citations
TL;DR: In this paper, it was shown that kinematic spaces with nontrivial dilatations are commutative, i.e., they are non-convex.
Abstract: In this paper we show that kinematic spaces with nontrivial dilatations are commutative.
15 citations
TL;DR: In this paper, the authors give an arithmetical characterization of graphs which are realizable as graphs of the lattice ℒ of the subspaces of a graphic (or projective) space of finite dimension and of finite order q ≥ 1.
Abstract: We give an arithmetical characterization of graphs which are realizable as graphs of the lattice ℒ of the subspaces of a graphic (or projective) space of finite dimension and of finite order q ≥ 1. In other words ℒ is any complemented modular lattice of finite rank and of finite order q. When q ≥ 2 and the rank is at least four, ℒ is the lattice of the subspaces of a finite dimensional vector space over the field GF(q). Two independent axioms are required involving the number of geodesies between any two vertices. This number must be a simple function of the distance.
11 citations
TL;DR: In this article, it was shown that the smallest cardinality of a blocking set is 2q−1 in any arbitrary affine plane αq (desarguesian or not) with q⩾5.
Abstract: Denote byαq an affine plane of order q. In the desarguesian case αq=AG(2,q), q ⩾ 5(q= ph, p prime), we prove that the smallest cardinality of a blocking set is 2q−1. In any arbitrary affine plane αq (desarguesian or not) with q⩾5, for any integer k with 2q−1⩽ k⩽(q−1)2, we construct a blocking set S with ¦S¦=k. For an irreducible blocking set S of αq we determine the upper bound S⩾ [q√q]+1. We prove that if ⇌q contains a blocking set S which is irreducible with its complementary blocking set, then necessarily αq=AG(2, 4) and S is uniquely determined. Finally we introduce techniques to obtain blocking sets in AG(2, q) and in PG(2, q).
9 citations
TL;DR: In this article, the authors define a distance d on the set of r-spaces of an n-space and transfer it to the GrasmannianG=G(n, r) to obtain a distinguished class of normal rational curves of order 1.
Abstract: We define a distance d on the set of r-spaces of an n-space. By the transfer of d to the GrasmannianG=G(n, r) we obtain a distinguished class of normal rational curves of order 1, the “1-distance lines’, 1=1,..., r, which are in 1–1-correspondence to the so-called “generalized reguli of type (r, 1)”.
9 citations
TL;DR: In this paper, it was shown that the image space may also be a three-dimensional flag space, and that the one-dimensional one-parameter motion in an isotropic plane can be interpreted in this flag space.
Abstract: The BLASCHKE — GRONWALD — map associates to each curve of a quasi elliptic space a planar euclidean one — parameter motion. K. STRBECKER has shown, that there exists an analog kinematic mapping for motions in an isotropic plane; image space is an isotropic space. Here it is proved, that this image space may also be a three — dimensional flag space. Considering the image curve of an one — parameter motion in an isotropic plane in this flag space we find an interpretation of all known properties of this motion.
7 citations
TL;DR: In this article, the ordering of an AH-plane is shown to induce an ordering on each coordinate biternary ring which is compatible with the multiplication of the associated algebra, and even stronger projective order relations are introduced and are also compatible with ternary operations.
Abstract: Preorderings and orderings of affine Hjelmslev planes were defined in [3]. The ordering of an AH-plane is shown to induce an ordering on each coordinate biternary ring which is compatible with the multiplication of the associated algebra. Even stronger projective order relations, on both the planes and biternary rings, are introduced and are shown to be also compatible with the ternary operations.
7 citations
5 citations
TL;DR: In this paper, a blocking set in the projective plane of order q was shown to exist for any integer k with q+m(q)+1 ⩽ k ⩾ q2−m(qs) having exactly k elements, where q denotes the greatest divisor of h different from h.
Abstract: Denote by PG(2,q) the finite desarguesian projective plane of order q, where q=ph, p a prime, q>2. We define the function m(q) as follows: m(q)=√q, if q is a square; m(q)=(q+1)/2, if q is a prime; m(q)=ph−d, if q=ph with h an odd integer, where d denotes the greatest divisor of h different from h. The following theorem is proved: For any integer k with q+m(q)+1 ⩽ k ⩽ q2−m(q), there exists a blocking set in PG(2,q) having exactly k elements.
TL;DR: In this paper, the following theorem is proved: consider K = GF(q), q e {32,64,81,128}, σ:K2 → K2 bijective such that σ is a semi-isometry.
Abstract: By using a computer the following theorem is proved: Consider K=GF(q), q e {32,64,81,128}, σ:K2 → K2 bijective such that
$$\overline {PQ} = 1$$
⇒
$$\overline {P^\sigma Q^\sigma } = 1$$
∀ P,Q e K2. Then σ is a semi-isometry. The assumption bijective can be dropped if q e {32,128}.
TL;DR: In this paper, the authors describe mappings of a plane K2 into itself such that P,Q ∈ K2 are linearly dependent only if P=Q+z,1/z, z ∈ k, z ≠ O.
Abstract: Dilatations of a plane K2 (K a field) are mappings such that\(\overrightarrow {PQ} \) and\(\overrightarrow {P^\sigma Q^\sigma } \)are linearly dependent vectors for all P,Q ∈ K2. In this note we describe mappings of K2 into itself such that\(\overrightarrow {PQ} \) and\(\overrightarrow {P^\sigma Q^\sigma } \)are linearly dependent only if P=Q+(z,1/z), z ∈ K, z ≠ O, by derivations of K. In finite planes we get dilatations as before.
TL;DR: In this article, the authors classify the finite linear spaces for which there is an integer d1 such that for any ordered pair (L,L′) of disjoint lines and for any point x outside L and L′, there are exactly d1 lines through x intersecting L but not L′.
Abstract: We classify the finite linear spaces for which there is an integer d1 such that for any ordered pair (L,L′) of disjoint lines and for any point x outside L and L′, there are exactly d1 lines through x intersecting L but not L′. Unless all lines have size 2, such a space is necessarily a semi-affine plane or a generalized projective 3-space.
TL;DR: In this paper, a new translation plane of order 25 is constructed, which has a collineation group acting on the line at infinity as a permutation group Z of order 48 with the properties:==================>>\s
Abstract: In this paper a new translation plane of order 25 is constructed. It has a collineation group acting on the line at infinity as a permutation group Z of order 48 with the properties:
(i)
Z contains a normal subgroup 1/2M of order 3 such that Z/1/2M is the direct product of an involution with a dihedral group of order 8.
(ii)
The orbits of Z have lengths 2, 12, 12.
TL;DR: In this article, Ribaucour et al. studied the properties of a Cω-representation of the spherical image of a regular surface Σ and gave characterizations for Φ being either a plane surface or a minimal surface.
Abstract: Let Φ be the middle surface of an isotropic rectilinear congruence Σ of class C3 in the real Euclidean space E3. When the spherical image of Σ is parametrized by special isothermal coordinates (u,v) eG ⊂ ℝ2, Φ can be described by a “generating” harmonic function A(u,v). Using such a Cω-representation of Σ, the basic properties of regularity and curvature of Φ are discussed. Moreover, the cases that Φ be a minimal (regular) surface Φ1, or a plane surface Φ2 are solved explicitly. In connection with the latter results (which are already well-known from Ribaucour) several new characterizations for Φ being a regular surface Φ1 resp. Φ2 are given: they are based on special properties (like: being asymptotic lines resp. lines of curvature of Φ) of those curves c (“Σ-Spurlinien”) in Φ the tangents of which form in each point Xec a minimal angle with the straight line of Σ passing through X.
TL;DR: In this article, the authors give a new proof of Nolte's result which does not depend on the characteristic of the underlying field and show that the generated protective Clifford group is isomorphic to the extended reflection group.
Abstract: The “extended reflection group” of a metric vector space was introduced by Nolte [10] to get a group theoretical representation of the corresponding protective metric geometry (in the sense of Schroder [12]). Nolte characterizes the extended. reflection, group among all representing groups if the characteristic of the underlying field is ≠ 2. We give a new proof of Nolte's result which does not depend on the characteristic. As a consequence we get that the “generated protective Clifford group” (see [12]) is isomorphic to the extended reflection group. Finally, we give examples of other representing groups.
TL;DR: In this paper, the minimum of generators with special geometrical properties for simply closed generalized ruled surfaces in Euclidean n-space was shown to be a vertex theorem.
Abstract: In Euclidean n-space we prove two vertex theorems for simply closed generalized ruled surfaces. The statements refer to the minimum of generators with special geometrical properties, and generalize theorems of SABAN[9] about ruled surfaces in Euclidean 3-space. Two examples are given.
TL;DR: In this article, the authors classified all Dembowski semi-i-spaces, i⩾3, into three classes: (1) the semi-planes which were determined by P.
Abstract: The Dembowski semi-planes are the semi-planes which were determined by P. Dembowski [1]. A Dembowski semi-i-space (i⩾1) is an incidence structure J=(P,B,I) for which: (i) each element of B is incident with at least i+3 elements of P, and (ii) each i-residual space of J is a Dembowski semi-plane. The article [4] contained the complete classification of all Dembowski semi-2-spaces, in this article we classify all Dembowski semi-i-spaces, i⩾3.
TL;DR: The concept of curvature χ was introduced by W. Benz as mentioned in this paper, who showed that among regular curves in\(\mathfrak{L}\) the chains of Σ(ℜ, Π) are characterized by χ = 0.
Abstract: Lie theory renders a concept of curvature χ for the geometries Σ(ℜ,ϰ) introduced by W. Benz [1]. Among regular curves in\(\mathfrak{L}\) the chains of Σ(ℜ,\(\mathfrak{L}\)) are characterized by χ=0.
TL;DR: In this article, an orthogonal decomposition of the corresponding metric vector space V4(K,f) into α-invariant subspaces Ui such that dim Ui ≤ 3 is given.
Abstract: If K is an Euclidean field and f a symmetric bilinear form of index 1, there is for each element α e O4 (K,f) an orthogonal decomposition of the corresponding metric vector space V4(K,f) into α — invariant subspaces Ui such that dim Ui ≤ 3. Thus it is possible to determine nomal form for α and to decompose the characteristic polynomial Xα into irreducible factors.
TL;DR: In this paper, 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.
Abstract: 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.
TL;DR: In this paper, the affine space of a Galois field over the prime field was analyzed and the following results were obtained: 1. AutC is isomorphic to the product of the augmented group of similarities (generated by similarities, quasi reflections, quasi rotations) and the group of collineations which are induced by the automorphism of GF(q) operating on the coordinates.
Abstract: In the affine plane over a Galois field GF(q), q ≡; 3(4), q = pα, of congruence transformations, of motions and of the generation of all point reflections respectively. Then we determine the groups AutC, AutM, AutM′ and obtain the following results: 1. Aut C is isomorphic to the product of the augmented group of similarities (generated by similarities, “quasi reflections”, “quasi rotations” 2) and the group of collineations which are induced by the automorphism of GF(q) operating on the coordinates. 2. AutM− AutC. 3. AutM′− group of affinities of the affine space of dimension 2α over the prime field. 4. Moreover for any desarguesian affine plane Aut Dil (Dil = group of dilatations) is isomorphic to Γ (the full collineation group).
TL;DR: In this paper, the triviality of the center of a group of motion in a metric geometry in the sense of Ewald was shown to imply a condition that admits a geometric interpretation.
Abstract: Let G be a group of motion in a metric geometry in the sense of Ewald. We give a condition (admitting a geometric interpretation) which implies the triviality of the center of G.
TL;DR: In this article, the authors studied the bundles of type 1, 2-called cyclic bundle and 38-called Strubecker-bundle and gave only the fundamental theorem and one metric result.
Abstract: According to K. Strubecker ([5]–[8]) a three dimensional real affine space with the metric ds2=dx2+dy2 is called a simply iso-tropic space J3(1). In J3(1) exist 41 types of bundles of linear line complexes. In this paper we study the bundles of type 1, 2-called cyclic bundle and 38-called Strubecker-bundle. For a cyclic bundle and a Strubecker-bundle we give only the fundamental theorem and one metric result, however for a cyclic bundle the metric theory is completely developed.