Related papers: A common axiomatic basis for projective geometry a…
The purpose of this paper is to present projective geometry in a synthetic, visual and intuitive style through the central notion of harmonicity which leads to harmonic curves. This presentation includes new results, unpublished proofs of…
The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. The topics include Desargues's Theorem, harmonic conjugates, projectivities, involutions, conics, Pascal's Theorem,…
By providing explicit definitions, we show that in both affine and projective geometry of dimension $\geq 3$, considered as first-order theories axiomatized in terms of lines as the only variables, and the binary line-intersection predicate…
Problems related to projections on closed convex cones are frequently encountered in optimization theory and related fields. To study these problems, various unifying ideas have been introduced, including asymmetric vector-valued norms and…
Let M be a family of sequences (a_1,...,a_p) where each a_k is a flat in a projective geometry of rank n (dimension n-1) and order q, and the sum of ranks, r(a_1) + ... + r(a_p), equals the rank of the join a_1 v ... v a_p. We prove upper…
Of the great theories of classical mathematics, projective geometry, with its powerful concepts of symmetry and duality, has been exceptional in continuing to intrigue investigators. The challenge put forth by Errett Bishop (1928-1983),…
This article provides a new perspective on the geometry of a projective line, which helps clarify and illuminate some classical results about projective plane. As part of the same train of ideas, the article also provides a proof of the…
The space of n (ordered) points on the projective line, modulo automorphisms of the line, is one of the most important and classical examples of an invariant theory quotient, and is one of the first examples given in any course. Generators…
In this paper we give a short and elementary proof of a characterization of the projective space in terms of rational curves which was conjectured by Mori and Mukai. A proof was first given in a preprint by K. Cho, Y. Miyaoka and N.…
Many combinatorial properties of a point set in the plane are determined by the set of possible partitions of the point set by a line. Their essential combinatorial properties are well captured by the axioms of oriented matroids. In fact,…
One dimensional metrical geometry may be developed in either an affine or projective setting over a general field using only algebraic ideas and quadratic forms. Some basic results of universal geometry are already present in this…
Projection maps which appear in the theory of buildings and oriented matroids are closely related to the notion of shellability. This was first observed by Bj{\"o}rner. In this paper, we give an axiomatic treatment of either concept and…
A zone diagram is a relatively new concept which was first defined and studied by T. Asano, J. Matousek and T. Tokuyama. It can be interpreted as a state of equilibrium between several mutually hostile kingdoms. Formally, it is a fixed…
Several tools have been developed to enhance automation of theorem proving in the 2D plane. However, in 3D, only a few approaches have been studied, and to our knowledge, nothing has been done in higher dimensions. In this paper, we present…
The paper introduces the class of O-metric spaces, a novel generalization of metric-type spaces, classifying almost all possible metric types into upward and downward O-metrics. We list some topologies arising from O-metrics and discuss…
We established a fixed-point theorem for mapping satisfying a general contractive inequality of integral type depended an another function. This theorem substantially extend the theorem due to Branciari (2003) and Rhoades (2003)
Projective connections first appeared in Cartan's papers in the 1920's. Since then they have resurfaced periodically in, for example, integrable systems and perhaps most recently in the context of so called projectively equivariant…
We consider moduli spaces of dynamical systems of correspondences over the projective line as a generalization of moduli spaces of dynamical systems of endomorphisms on the projective line. We obtain the rationality of the moduli spaces.…
We provide a variant of an axiomatization of elementary geometry based on logical axioms in the spirit of Huzita--Justin axioms for the Origami constructions. We isolate the fragments corresponding to natural classes of Origami…
We explore the concept of projections of syzygies and prove two new technical results; we firstly give a precise characterization of syzygy schemes in terms of their projections, secondly, we prove a converse to Aprodu's Projection Theorem.…