Related papers: Bruck nets and partial Sherk planes
This paper introduces ordered skew fields that result from the construction of a skew field over an ordered line in a Desargues affine plane. A special case of a finite ordered skew field in the construction of a skew field over an ordered…
In this paper, we consider point sets of finite Desarguesian planes whose multisets of intersection numbers with lines is the same for all but one exceptional parallel class of lines. We call such sets regular of affine type. When the lines…
This paper introduces advances in the geometry of the ratio of either two or three points in a line in the Desargues affine plane, and we see this as a ratio of elements of skew field which are constructed over a line in Desargues affine…
The paper is devoted to a detailed self-contained exposition of a part of the theory of affine planes leading to a construction of affine (or, equivalently, projective) planes not satisfying the Desarques axiom. It is intended to complement…
We recall the well-known Chern-Terng theorem concerning affine minimal surfaces. Next we formulate some complementary (with transversal fields necessarily not parallel) affine B\"acklund theorem. We describe some geometrical conditions…
The fundamental theorem of affine geometry says that a self-bijection $f$ of a finite-dimensional affine space over a possibly skew field takes left affine subspaces to left affine subspaces of the same dimension, then $f$ of the expected…
The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine. In this note we prove…
We prove that if $C$ is a reflexive smooth plane curve of degree $d$ defined over a finite field $\mathbb{F}_q$ with $d\leq q+1$, then there is an $\mathbb{F}_q$-line $L$ that intersects $C$ transversely. We also prove the same result for…
The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study…
Affine rotation surfaces are a generalization of the well-known surfaces of revolution. Affine rotation surfaces arise naturally within the framework of affine differential geometry, a field started by Blaschke in the first decades of the…
The symmetries described by Pin groups are the result of combining a finite number of discrete reflections in (hyper)planes. The current work shows how an analysis using geometric algebra provides a picture complementary to that of the…
The paper reports the progress with the classical problem, posed by Blaschke and Bol in 1938. We present new examples and new classifications of natural classes of hexagonal circular 3-webs. The main results is the classification of…
We classify nets of conics in Desarguesian projective planes over finite fields of odd order, namely, two-dimensional linear systems of conics containing a repeated line. Our proof is geometric in the sense that we solve the equivalent…
By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are…
An associative division algebra D is said to be _affine_ over a central subfield k if D is finitely generated as a k-algebra. In 1956 Amitsur famously proved that, when k is uncountable, D cannot be k-affine unless D is algebraic over k. In…
We prove that affine maps are uniquely extremal quasiconformal maps on the complement of a well distribute set in the complex plane answering a conjecture from \cite{markovic}. We construct the required Reich sequence using Bergman…
Let $T$ be a triangulation of a Riemann surface. We show that the 1-skeleton of $T$ may be oriented so that there is a global bound on the outdegree of the vertices. Our application is to construct extremal metrics on triangulations formed…
It is well known that pseudo-Riemannian metrics in the projective class of a given torsion free affine connection can be obtained from (and are equivalent to) the solutions of a certain overdetermined projectively invariant differential…
Based on an ordering with directed lines and using constructions instead of existential axioms, von Plato proposed a constructive axiomatization of the ordered affine geometry. There are 22 axioms for the ordered affine geometry, of which…
The degenerate affine and affine BMW algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic Lie algebras and quantum groups, respectively. Cyclotomic BMW algebras, affine Hecke algebras, cyclotomic Hecke…