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Showing papers in "Journal of Geometry in 1989"




Journal ArticleDOI
TL;DR: In this article, the authors characterize subsets of incidence spaces, which are locally complete resp. bundletrue (bundeltreu), and specify necessary and sufficient conditions to extend the embedding of a subset of a space into another space to an embedding.
Abstract: We characterize subsets of incidence spaces, which are locally complete resp. bundletrue (bundeltreu). Then we specify necessary and sufficient conditions to extend the embedding of a subset of a space into another space to an embedding of the complete space. Using these results we show that every ordered space (not necesserily fulfilling (O3), cf. §4) can be embedded into a projective space. The last result is a generalization of former results using the additional assumption (O3) (cf. [2,9,10]).

14 citations


Journal ArticleDOI
Dongming Wang1
TL;DR: In this article, Wu's theorem prover was used to prove that the six intersection points of every Pappus line and its corresponding Leisenring line are collinear.
Abstract: Concerning Pappus lines and Leisenring lines there is a set of interesting theorems given in [1]. In this paper, we show that these theorems can be easily proved by our computer prover which is implemented on the basis of Wu's method for mechanical theorem proving in geometries. Using this prover we discovered, furthermore, a new theorem: The six intersection points of every Pappus line and its corresponding Leisenring line are collinear.

14 citations


Journal ArticleDOI
TL;DR: In this article, a finite affine plane that permits derivation and a derivable net embedded in it is considered and the main question with regard to the collineation group of is whether the full group of the derived plane is the inherited group.
Abstract: Let ~r be a finite affine plane that permits derivation and let R denote a derivable net embedded in r. The collineation group of the derived plane r obtained by deriving R which leaves the replaced net/~ invariant induces a collineation group in r and which is called the inherited group. The main question with regard to the collineation group of is whether the full group of ~ is the inherited group.

12 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that a spread S over a locally compact non-screte field F defines a topological translation plane if and only if the spread is compact.
Abstract: We prove that a spread S over a locally compact nondlscrete field F defines a topological translation plane if and only if the spread is compact. For F=R, this is implicit in Breuning's thesis [Bre], cf. [B 2]. For the proof, we describe the point set of the projective translation plane as a quotient space of some projective space, with identifications taking place in one hyperplane. This is new even for F=R.

12 citations



Journal ArticleDOI
TL;DR: In this paper, it was shown that a closed convex polygonal curve on the surface of a 3-polytope develops in the plane to a simple path: it does not self-intersect.
Abstract: It is shown that a closed convex polygonal curve on the surface of a 3-polytope develops in the plane to a simple path: it does not self-intersect.

10 citations


Journal ArticleDOI
TL;DR: In this paper, the problem of determining all the triples of circle-pencils forming a hexagonal 3-web is completely solved, which is the state-of-the-art.
Abstract: In this paper, the problem of determining of all the triples of circle-pencils forming a hexagonal 3-web is completely solved.

9 citations


Journal ArticleDOI
Peter Massopust1
TL;DR: In this article, the theory of iterated function systems (i.f.s) is used to construct and geometrically describe Peano curves, whose attractors are the graphs of some well-known Peano curve graphs.
Abstract: We show that the theory of iterated function systems (i.f.s.′s) can be used to construct and geometrically describe Peano curves. We present this point of view by exhibiting i.f.s.′s whose attractors are the graphs of some well-known Peano curves.

8 citations


Journal ArticleDOI
TL;DR: In this article, an extension of the traditional geometry of the triangle is derived through the construction of two 30-points particular cubics, and two generation procedures founded on triangular quadratic transformations and dual associate properties of the two cubics are presented.
Abstract: An extension of the traditional geometry of the triangle is derived through the construction of two 30-points particular cubics. Two generation procedures founded on triangular quadratic transformations and dual associate properties of the two cubics are presented.

Journal ArticleDOI
TL;DR: In this article, the unifying framework of families of convexity spaces for the treatment of various notions of planar convexness and the associated convex hulls is utilized.
Abstract: We utilize the unifying framework of families of convexity spaces for the treatment of various notions of planar convexity and the associated convex hulls. Our major goal is to prove the refinement and decomposition theorems for families of convexity spaces. These general theorems are then applied to two examples: restricted-oriented convex sets andNESW-convex sets. The applications demonstrate the usefulness of these general theorems, since they give rise to simple algorithms for the computation of the associated convex hulls of polygons.

Journal ArticleDOI
TL;DR: In this article, a classification and determination of all porous double spaces which are embedable in a projective space and where each line is incident with at least five points is given.
Abstract: A classification and determination of all porous double spaces which are embedable in a projective space and where each line is incident with at least five points is given. The main results are summed up in the theorem at the end of the introduction.

Journal ArticleDOI
TL;DR: A characterization for chordal graphs which are join spaces is given in this paper. But this characterization assumes that a chordal graph is a join space if and only if it does not contain one of the two forbidden graphs as an induced subgraph.
Abstract: A join space is an abstract model for partially ordered linear, spherical and projective geometries. A characterization for chordal graphs which are join spaces is given: a chordal graph is a join space if and only if it does not contain one of the two forbidden graphs as an induced subgraph.

Journal ArticleDOI
TL;DR: A pseudo-Riemannian surface M in a pseudo-Euclidean space Esm is said to have planar normal sections if normal sections of M are planar curves as discussed by the authors.
Abstract: A pseudo-Riemannian surface M in a pseudo-Euclidean space Esm is said to have planar normal sections if normal sections of M are planar curves. In the present paper we give some classification theorem concerning surfaces in Ems with planar normal sections.

Journal ArticleDOI
TL;DR: In this article, the existence of blocking sets in a particular class of Sλ (3,4,v) was characterized and a construction to obtain Sλ(3, 4,2v) having blocking sets was given.
Abstract: We characterize the existence of blocking sets in a particular class of Sλ (3,4,v) and we give a construction to obtain Sλ(3,4,2v) having blocking sets.

Journal ArticleDOI
TL;DR: The orthomorphism graph of Zp, denoted Orth(Zp), has as its vertex set the orthomorphisms of Z p as mentioned in this paper, two of which are adjacent if the mapping 6:Z -.Z defined by 6 (x)=~(x) #(x)) is a bijection.
Abstract: An orthomorphism of Z , p an odd prime, is a bijection @:Z -~Z o ~(0)=0, such P P p that the mapping 9:Zp-~ZD defined by U(x)=@(x)-x is also a bijection. The orthomorphism graph of Zp, denoted Orth(Zp), has as its vertex set the orthomorphisms of Zp, two orthomorphisms #,# of Zp being adjacent if the mapping 6:Z -.Z defined by 6(x)=~(x) #(x) is a bijection. For the P P definition of and known results on orthomorphisms of groups in general see Johnson, Dulmage and Mendelsohn [9] or D~nes and Keedwell [4,chapter 7]. Assuming Z to be the additive group of GF(p) enables us to use P multiplication in GF(p) to define classes of orthomorphisms and to study adjacencies within these classes. As an example, if we use I to denote the a mapping x-~ax, then clearly IaeOrth(Z p) if and only if a#O, i and !a is adjacent to I b if and only if a#bo We use S 1 to denote the set {I~; a#O,l}. Another class of mappings that give rise to orthomorphisms can be defined as follows:


Journal ArticleDOI
TL;DR: In this article, it was shown that the results of BICHARA-KorCHMAROS [1] and WETTL [16] cannot be extended to non-desarguesian planes.
Abstract: In this paper we give k-sets (k>-q+1) of the Hall planes with q-√q nuclei. This shows that the results of BICHARA — KORCHMAROS [1] and WETTL [16] cannot be extended to non-desarguesian planes. We construct some inherited arcs in the Hall planes as well. For example complete (q−1)-arcs of the Hall planes of odd order will be constructed.

Journal ArticleDOI
TL;DR: In this paper, a characterization theorem is established: SetS is a compact union of two starshaped sets if and only if there is a sequence of polytopes converging to S such that each set satisfies propertyK ≥ 2.
Abstract: SetS inR d has propertyK 2 if and only ifS is a finite union ofd-polytopes and for every finite setF in bdryS there exist points c1,c2 (depending onF) such that each point ofF is clearly visible viaS from at least one ci,i = 1,2. The following characterization theorem is established: Let $$S \subseteq R^d$$ , d≠2. SetS is a compact union of two starshaped sets if and only if there is a sequence {S j } converging toS (relative to the Hausdorff metric) such that each setS j satisfies propertyK 2. For $$S \subseteq R^2$$ , the sufficiency of the condition above still holds, although the necessity fails.

Journal ArticleDOI
TL;DR: In this article, the atoms of the lattice of compatible topologies on a given projective plane are examined, and the notion of a field of type V is generalized to ternary fields of V, which arise for example as coordinate structure of strictly uniformizable or of orderable topological planes.
Abstract: The atoms of the lattice of compatible topologies on a given projective plane are examined. The notion of a field of type V is generalized to ternary fields of type V. These are always minimal, and they arise for example as coordinate structure of strictly uniformizable or of orderable topological planes. Hence, such planes are minimal.

Journal ArticleDOI
TL;DR: In this paper, it was shown that a 4-point subset of a set of positive two-dimensional Lebesgue measures has a common point if and only if every finite subset of the set sees viaS such a set.
Abstract: Let\(S \subseteq R^2\) and assume that there is a countable collection of lines {L i : 1 ≤ i} such that (int cl S)\( \sim S \subseteq \cup \{ L_i :1 \leqslant i\}\) and ((int cl S) ∼S) ∩Li has one-dimensional Lebesgue measure zero, 1 ≤i. Then every 4 point subset ofS sees viaS a set of positive two-dimensional Lebesgue measure if and only if every finite subset ofS sees viaS such a set. Furthermore, a parallel result holds with ‘two-dimensional’ replaced by ‘one-dimensional’. Finally, setS is finitely starlike if and only if every 5 points ofS see viaS a common point. In each case, the number 4 or 5 is best possible.

Journal ArticleDOI
TL;DR: Among the known finite Minkowski planes, a family of examples admits a partition of the set of blocks into equivalence classes, each of which in turn partitions the point set.
Abstract: Among the known finite Minkowski planes we determine an infinite family of examples admitting a partition of the set of blocks into equivalence classes, each of which in turn partitions the point set; in particular non-miquelian finite Minkowski planes with this property exist.

Journal ArticleDOI
TL;DR: In this article, a construction of (s,k,λ)-partitions of finite non-abelian p-groups and of Frobenius groups with nonabelian kernel is given.
Abstract: For λ >1 and many values of s andk, we give a construction of (s,k,λ)-partitions of finite non-abelian p-groups and of Frobenius groups with non-abelian kernel. These groups are associated with translation transversl designs of the same parameters.

Journal ArticleDOI
TL;DR: In this paper, the authors give a characterization of topological affine spaces by means of the topologies on the sets of points and lines, where the joining of points, the intersection of n independent hyperplanes and the construction of parallel hyperplanes are continuous operations.
Abstract: Some characterizations of the topological affine spaces are already known [2,5,6]; they are given via the topologies on the sets of points and hyperplanes. According to the definition made by Sorensen in [6], a topological affine space is an affine space whose sets of points and hyperplanes are endowed with non-trivial topologies such that the joining of n independent points, the intersection of n independent hyperplanes and the construction of parallel hyperplanes are continuous operations. In this paper we give a new characterization of such spaces by means of the topologies on the sets of points and lines.


Journal ArticleDOI

Journal ArticleDOI
TL;DR: In this paper, an infinite set of one-parameter motions generating a Steiner surface as path of a circle is given, among these motions are Darboux-motions, whereby the points move along congruent circles.
Abstract: Algebraic surfaces of fourth order containing three double lines with a common point are called Steiner-surfaces. These surfaces ψ contain a two-parameter set of conics lying in the tangent planes of ψ. According to WUNDERLICH [17] a Steiner-surface can be generated by translation of a parabola along a parabola if and only if two or three of the double lines coincide. If such a special double line, the tangent plane along it und the singular point lying on it are choosen to represent the absolute line, plane and point respectivly of a flag space F3, the conies of ψ are circles in the sense of F3. An infinite set of oneparameter motions generating ψ as path of a circle is given. Among these motions exist Darboux-motions, whereby the points move along congruent circles.