Related papers: Paths of rectangles inscribed in lines over fields

We develop a circle of ideas involving pairs of lines in the plane, intersections of hyperbolically rotated elliptical cones and the locus of the centers of rectangles inscribed in lines in the plane.

A parallelogram is conformally inscribed in four lines in the plane if it is inscribed in a scaled copy of the configuration of four lines. We describe the geometry of the three-dimensional Euclidean space whose points are the…

Binary relations are one of the standard ways to encode, characterise and reason about graphs. Relation algebras provide equational axioms for a large fragment of the calculus of binary relations. Although relations are standard tools in…

An arithmetical structure on a graph is given by a labeling of the vertices which satisfies certain divisibility properties. In this note, we look at several families of graphs and attempt to give counts on the number of arithmetical…

We prove an integral formula for continuous paths of rectangles inscribed in a piecewise smooth loop. We then use this integral formula to show that (with a very mild genericity hypothesis) the number of rectangle coincidences, informally…

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…

We study the topological configurations of the lines of principal curvature, the asymptotic and characteristic curves on a cuspidal edge, in the domain of a parametrization of this surface as well as on the surface itself. Such…

A projective rectangle is like a projective plane that has different lengths in two directions. We develop the basic theory of projective rectangles including incidence properties, projective subplanes, configuration counts, a partial…

Linear Geometry studies geometric properties which can be expressed via the notion of a line. All information about lines is encoded in a ternary relation called a line relation. A set endowed with a line relation is called a liner. So,…

A projective rectangle is like a projective plane that may have different lengths in two directions. We develop properties of the graph of lines, in which adjacency means having a common point, especially its strong regularity and clique…

We derive the equations of chains for path geometries on surfaces by solving the equivalence problem of a related structure: sub-Riemannian geometry of signature $(1,1)$ on a contact 3-manifold. This approach is significantly simpler than…

Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve have attracted much interest. In the present paper, we propose a new method to construct a surface…

We apply an old method for constructing points-and-lines configurations in the plane to study some recent questions in incidence geometry.

In the first part of this article, we consider ruled surfaces defined over a finite field; we introduce invariants for them, and describe some explicit contructions that illustrate possible behaviour of these invariants. In the second part,…

We discuss the principle tools and results and state a few open problems concerning the classification and topology of plane sextics and trigonal curves in ruled surfaces.

We study cubic rational maps that take lines to plane curves. A complete description of such cubic rational maps concludes the classification of all planarizations, i.e., maps taking lines to plane curves.

We give geometric ways to determine the cross ratio of conics in a pencil using elementary projective geometry.

We study the geometry of spaces of planes on smooth complete intersections of three quadrics, with a view toward rationality questions.

Stroking a path is one of the two basic rendering operations in vector graphics standards (e.g., PostScript, PDF, SVG). We survey path stroking rendering results from real-world software implementations of path stroking for anecdotal…

Given a finite collection of two-dimensional tile types, the field of study concerned with covering the plane with tiles of these types exclusively has a long history, having enjoyed great prominence in the last six to seven decades. Much…