Showing papers in "Journal of Geometry in 1975"
TL;DR: In this article, the maximum number of mutually disjoint S(λ;2,3,v) for all v ≡ 0 or 4 (mod 6) and v ≡ 2 (mod 2) was proved.
Abstract: Let D(λ;v) denote the maximum number of mutually disjoint S(λ;2,3,v). We prove that D(2;v)=v−2/2 for all v ≡ 0 or 4 (mod 6), v ≡ 0 and that D(6;v)=v−2/6 for all v ≡ 2 (mod 6).
26 citations
TL;DR: In this article, it was shown that all affine ring geometries over a Z-ring are affine Barbilian planes and represent their automorphisms algebraically as semilinear bijections.
Abstract: In part I we proved that every affine Barbilian plane is up to isomorphisms an affine geometry over a Z-ring R as defined in the introducing abstract there. Now we carry out that conversely all affine ring geometries over a Z-ring are affine Barbilian planes and represent their automorphisms algebraically as semilinear bijections. Finally, we present several classification theorems as, for instance, that the class of Desarguesian affine planes coinzides with the class of affine Barbilian planes, satisfying the additional axiom that two different points are always non-neighboured. The weaker condition that there is always exactly one line passing through two different points corresponds with the fact that the underlying ring is a right Bezoutring. There is at most one line passing through two different points iff the corresponding ring R has no zero-divisors.
18 citations
TL;DR: In this paper, the authors give a synthetic proof of the same result for any euclidean plane, and they do not assume the axiom of completeness, but they do assume that the real field preserves a non-zero distance.
Abstract: In [1,5 ] the authors prove that any transformation of the euclidean plane,coordinatized by the real field, which preserves a single non-zero distance is necessarily an isometry. The purpose of this note is to give a synthetic proof of the same result forany euclidean plane. Hence, I do not assume the axiom of completeness.
9 citations
TL;DR: In the Euclidean geometry of the triangle, the intersections of internal and external angle bisectors with opposite sides, and the feet of altitudes give rise to sixteen Graves triads, in four sets of four each.
Abstract: A Graves triad is a cyclic triad of triangles, each circumscribing the next, thus forming a Pappus configuration. In the Euclidean geometry of the triangle the points of contact of the inscribed and escribed circles, the intersections of internal and external angle bisectors with opposite sides, and the feet of altitudes give rise to sixteen Graves triads, in four sets of four each. Each set of four is associated with a rectangular hyperbola through the vertices, orthocentre, in-centre, Gergonne point, and Nagel point, or their external analogues, and having its centre at a Feuerbach point.
7 citations
5 citations
TL;DR: The concept of a generalized affine plane was introduced in this article, where a group of (bijective) dilations and a subgroup of translations give a nearring of trace preserving quasi-endomorphisms and there is a sub group fo the translations, called the semi-identities, that give an ideal in this near-ring.
Abstract: The concept of a generalized affine plane was introduced in [2]. To each such generalized affine plane there is a group of (bijective) dilations and a subgroup of translations. The subgroup of translations gives a nearring of trace preserving quasi-endomorphisms and there is a subgroup fo the translations, called the semi-identities, that give an ideal in this near-ring. The quotient nearring is a skew field.
4 citations
3 citations
1 citations
[...]
TL;DR: In this article, a common foundation is given for K. Vitzthum's [4], V. Groze's [1] and R.Sandler's [3] generalisations of the concept of projective plane.
Abstract: In this paper a common foundation is given for K. Vitzthum's [4], V. Groze's [1] and R.Sandler's [3] generalisations of the concept of projective (resp.affine) plane.Particular features of the translation planes are studied.
1 citations
TL;DR: In this paper, a syntactical "princple" similar to the usual duality principle is developed which relates the properties of a Baer subplane to its relative complement.
Abstract: A syntactical “princple” similar to the usual Duality Principle is developed which relates the properties of a Baer subplane to the properties of the subplane's relative complement. (In fact this principle is shown to characterize the notion of Baer subplane.) Two known results are presented (and extended) from this point-of-view.
TL;DR: In this article, it was shown that any 4 tangents of C are met by exactly 2 lines in real projective 3-spaceP3, where P3 is a projective curve of order 3.
Abstract: Let C be a directly differentiable curve of order 3 in real projective 3-spaceP3; cf. 3. It is known that there exist 4 tangents of C which are met by a line; cf. 3. The purpose of this paper is to show that, in fact, any 4 tangents of C are met by exactly 2 lines; cf. Theorem 3.3.
TL;DR: In this paper, the self-dual sentences of projective geometry are characterized in terms of their logical structure, and a syntactical analysis reveals an unexpected relationship with graph theory, the self dual sentences being precisely the geometric interpretations (in a reasonably natural sense) of arbitrary sentences of graph theory.
Abstract: The self-dual sentences of projective geometry are characterized in terms of their logical structure. This syntactical analysis reveals an unexpected relationship with graph theory, the self-dual sentences being precisely the geometric interpretations (in a reasonably natural sense) of arbitrary sentences of graph theory.