On collinear Griffiths points

J. Tabov has proved [1] that four Griffiths points are collinear if the vertices of a given quadrangle are on a circle. In this article we prove some generalization of this result in a very simple geometrical way (based on Desargues theorem). Received 9 July 1999; revised 13 December 1999.     

共线点与共点线问题的研究

利用向量法、坐标法、仿射变换以及射影几何中的德萨格定理、帕斯卡定理和布利安桑定理,解决初等几何中的共线点和共点线问题.     

On a theorem about projectivities

The main result of this paper is a theorem about projectivities in then-dimensional complex projective spaceP ⁿ (n 2). Puttingn = 2, we showed in [3] that the theorem of Desargues inP ⁿ is a special case of this theorem. And not only the theorem of Desargues but also the converse of the theorem of Pascal, the theorem of Pappus-Pascal, the theorem of Miquel, the Newton line, the Brocard points and a lot of lesser known results in the projective, the affine and the Euchdean plane were obtained from this theorem as special cases without any further proof. Many extensions of classical theorems in the projective, affine and Euclidean plane to higher dimensions can be found in the literature and probably some of these are special cases of this theorem inP ⁿ. We only give a few examples and leave it as an open problem which other special cases could be found.     

Early explicit examples of non-desarguesian plane geometries

We trace the history of geometries where Desargues?? theorem is not valid. Roughly we cover the time from the middle of the nineteenth century until the first decade of the twentieth century, discussing work by Beltrami, Klein, Wiener, Peano, Moulton, and of course Hilbert.     

Projectivities between bundles of plane higher order curves

In [5] we give a theorem about projectivities between three line bundles in the projective plane with an almost trivial proof, and with a lot of special cases among which the theorems of Desargues, Pappus-Pascal, Pascal and many other lesser known results. In this paper we retake this idea, but now for projectivities between bundles of higher order curves in the plane.     

The maximal size of 6- and 7-arcs in projective Hjelmslev planes over chain rings of order 9

We complete the determination of the maximum sizes of (k,n)-arcs,n≤12,in the projective Hjelmslev planes over the two (proper) chain rings Z 9=Z/9Z and S 3=F 3 [X]/(X 2) of order 9 by resolving the hitherto open cases n=6 and n=7.Parts of our proofs rely on decidedly geometric properties of the planes such as Desargues’ theorem and the existence of certain subplanes.     

A case study in formalizing projective geometry in Coq: Desargues theorem

Formalizing geometry theorems in a proof assistant like Coq is challenging. As emphasized in the literature, the non-degeneracy conditions lead to long technical proofs. In addition, when considering higher-dimensions, the amount of incidence relations (e.g. point–line, point–plane, line–plane) induce numerous technical lemmas. In this article, we investigate formalizing projective plane geometry as well as projective space geometry. We mainly focus on one of the fundamental properties of the projective space, namely Desargues property. We formally prove that it is independent of projective plane geometry axioms but can be derived from Pappus property in a two-dimensional setting. Regarding at least three-dimensional projective geometry, we present an original approach based on the notion of rank which allows to describe incidence and non-incidence relations such as equality, collinearity and coplanarity homogeneously. This approach allows to carry out proofs in a more systematic way and was successfully used to fairly easily formalize Desargues theorem in Coq. This illustrates the power and efficiency of our approach (using only ranks) to prove properties of the projective space.     

用透视对应方法证明Desargues定理

用透视对应的方法给出射影几何中Desargues定理新的、更为简洁的证明.     

Eine Bemerkung zum grossen Satz von Desargues in affinen Hjelmslev-Ebenen

An affine Hjelmslev-plane is called desarguesian if and only if the translations form a transitive group and for each pair t₁ and t₂ of translations, where each trace of t₁ is a trace or t₂, there exists a trace preserving endomorphism, which maps t₁ in t₂. The purpose of this paper is to charakterize desarguesian affine Hjelmslev-planes by means of a condition, which corresponds to the theorem of Desargues in ordinary affine planes, including the case, that each line contains only three classes of neighbour points. This case was omitted in [5].     

Connections between projective geometry and superassociative algebra

This paper establishes a connection between projective geometry and the superassociative algebra of multiplace functions. In Part I a quaternary operation is defined for the points on a line j in a projective plane π relative to a fixed quadrangle and the result of operating on A,B,C,D is denoted by A(B,C,D). It is shown that π is a Pappus plane if and only if this operation is superassociative: (A(B,C,D))(E,F,G)=A(B(E,F,G),C(E,F,G),D(E,F,G)) for any points A,B,C,D,E,F,G on j. Desargues and certain special Desargues planes are also characterized by restrictions of the superassociative law. In Part II projective geometry is developed over superassociative systems.     

Ten Exceptional Geometries from Trivalent Distance Regular Graphs

The ten distance regular graphs of valency 3 and girth > 4 define ten non-isomorphic neighborhood geometries, amongst which a projective plane, a generalized quadrangle, two generalized hexagons, the tilde geometry, the Desargues configuration and the Pappus configuration. All these geometries are bislim, i.e., they have three points on each line and three lines through each point. We study properties of these geometries such as embedding rank, generating rank, representation in real spaces, alternative constructions. Our main result is a general construction method for homogeneous embeddings of flag transitive self-polar bislim geometries in real projective space.     

A generalized Desargues configuration and the pure braid group

In this paper, a configuration with n = (₂^d) points in the plane is described. This configuration, as a matroid, is a Desargues configuration if d = 5, and the union of (₅^d) such configurations if d> 5. As an oriented matroid, it is a rank 3 truncation of the directed complete graph on d vertices. From this fact, it follows from a version of the Lefschetz-Zariski theorem implied by results of Salvetti that the fundamental group π of the complexification of its line arrangement is Artin's pure (or coloured) braid group on d strands.

In this paper we obtain, by using techniques introduced by Salvetti, a new algorithm for finding a presentation of π based on this particular configuration.     

Projective-type axioms for the hyperbolic plane

It is shown that the Laws of Pappus and Desargues may replace the Axiom of Projectivities in Menger's development of hyperbolic geometry from axioms of alignment.     

Placement of the Desargues configuration on a cubic curve

Hilbert and Cohn-Vossen once declared that the configurations of Desargues and Pappus are by far the most important projective configurations. These two are very similar in many respects: both are regular and self-dual, both could be constructed with ruler alone and hence exist over the rational plane, the final collinearity in both instances are automatic and both could be regarded as self-inscribed and self-circumscribed p9lygons (see [1, p. 128]). Nevertheless, there is one fundamental difference between these two configurations, viz. while the Pappus-Brianchon configuration can be realized as nine points on a non-singular cubic curve over the complex plane (in doubly infinite ways), it is impossible to get such a representation for the Desargues configuration. In fact, the configuration of Desargues can be placed in a projective plane in such a way that its vertices lie on a cubic curve over a field k if and only if k is of characteristic 2 and has at least 16 elements. Moreover, any cubic curve containing the vertices of this configuration must be singular.This research of all the three authors was supported by the HSERC of Canada.     

Martingale convergence,series expansion of Gaussian elements,and strong law of large numbers in Fréchet spaces

The main result of this paper is the derivation of a convergence theorem for certain martingales with values in a separable Fréchet space F. It is shown that this result includes a well known theorem due to Chatterji. Moreover, the series expansion of zero-mean Gaussian elements with values in F and the strong law of large numbers for i.i.d. F-valued random elements also follow as applications of the main theorem.     

Nonconvex Separation Theorem for Multifunctions,Subdifferential Calculus and Applications

We derive a nonconvex separation theorem for multifunctions that generalizes an early result of Borwein and Jofré and show that this result is equivalent to several other subdifferential calculus results in smooth Banach spaces. Then we apply this nonconvex separation theorem to improve a second welfare theorem in economics and a necessary optimality condition for a multi-objective optimization problem.     

Product formulas,nonlinear semigroups,and accretive operators

A special case of our main theorem, when combined with a known result of Brezis and Pazy, shows that in reflexive Banach spaces with a uniformly Gâteaux differentiable norm, resolvent consistency is equivalent to convergence for nonlinear contractive algorithms. (The linear case is due to Chernoff.) The proof uses ideas of Crandall, Liggett, and Baillon. Other applications of our theorem include results concerning the generation of nonlinear semigroups (e.g., a nonlinear Hille-Yosida theorem for “nice” Banach spaces that includes the familiar Hilbert space result), the geometry of Banach spaces, extensions of accretive operators, invariance criteria, and the asymptotic behavior of nonlinear semigroups and resolvents. The equivalence between resolvent consistency and convergence for nonlinear contractive algorithms seems to be new even in Hilbert space. Our nonlinear Hille-Yosida theorem is the first of its kind outside Hilbert space. It establishes a biunique correspondence between m-accretive operators and semigroups on nonexpansive retracts of “nice” Banach spaces and provides affirmative answers to two questions of Kato.     

Modelling of Ring Geometry from von Neumann’s Point of View

The idea to consider different unions of points, lines, planes, etc., is rather old. Many important configurations of such kinds are geometric (or matroidal) lattices. In this work, we study Desargues, Pappus, and Pasch configurations in D-semimodular lattices.     

Weighted Means, Strict Ergodicity, and Uniform Distributions

We strengthen the well-known Oxtoby theorem for strictly ergodic transformations by replacing the standard Cesaro convergence by the weaker Riesz or Voronoi convergence with monotonically increasing or decreasing weight coefficients. This general result allows, in particular, to strengthen the classical Weyl theorem on the uniform distribution of fractional parts of polynomials with irrational coefficients.     

About a minimax theorem of Matthies,Strang and Christiansen

We give a new minisup theorem for noncompact strategy sets. Our result is of the type of the Matthies-Strang-Christiansen minimax theorem where the hyperplane should be replaced by any closed convex set. As an application, we derive a slight generalization of the Matthies-Strang-Christiansen minimax theorem.