Related papers: Projective Line Revisited
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…
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…
We determine those maps between affine or projective spaces that are linear in the abstract sense of transforming collinear points into collinear points and whose restriction to any line is constant or injective. Our results are extensions…
The coordinate projective line over a field is seen as a groupoid with a further `projection' structure. We investigate conversely to what extent such an, abstractly given, groupoid may be coordinatized by a suitable field constructed out…
A recent paper showed how to find sets of finite affine or projective planes constructed on a common set of points, so that lines of one plane meet lines of a different plane in at most two points. In this paper, those results are…
The purpose of the present article is to examine the essence of what has commonlybeen described as a "projective line", but which is here named a "meridian". This shall be done in several papers: this first paper devoted to the meridian…
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…
A projective link is a smooth closed 1-submanifold of the real projective space of dimension three. A projective link is said to be affine if it is isotopic to a link, which does not intersect some projective plane. The main result: a…
We describe a notion of (abstract) projective line over a field as a set equipped with a certain first order structure, and a projectivity between projective lines as a bijection preserving this structure. The structure in question is that…
Congruences, or $2$-parameter families of lines in $3$-space are of interest in many situations, in particular in geometric optics. In this paper we consider elements of their geometry which are invariant under affine changes of…
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…
A standard procedure in classical projective geometry, using pencils of lines to extend an incidence plane to a projective plane, is examined from a constructive viewpoint. Brouwerian counterexamples reveal the limitations of traditional…
The projective line over a field carries structure of a groupoid with a certain correspondence between objects and arrows. We discuss to what extent the field can be reconstructed from the groupoid.
The first part of this note contains a review of basic properties of the variety of lines contained in an embedded projective variety and passing through a general point. In particular we provide a detailed proof that for varieties defined…
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…
We offer an axiomatic presentation of three-dimensional projective space that adopts the line as its fundamental element and renders automatic the principle of duality.
Let k be a perfect field and let K/k be a finite extension of fields. An arithmetic noncommutative projective line is a noncommutative space equal to the projectivization of the noncommutative symmetric algebra of a k-central two -sided…
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…
We introduce an alternative formalization of curved spaces in which the concept of a pointwise affine space, as defined here, replaces that of a manifold. New or modified definitions of familiar notions from differential geometry such as…
This article provides a geometric representation for the well-known isomorphism between the special orthogonal group of an isotropic quadratic space of dimension 3 and the group of projective transformations of a projective line. This…