Axioms Of Incidence Research Articles

Incidence Axioms for the Boundary at Infinity of Complex Hyperbolic Spaces

Abstract We characterize the boundary at infinity of a complex hyperbolic space as a compact Ptolemy space that satisfies four incidence axioms.

Proving and Generalizing Desargues’ Two-Triangle Theorem in 3-Dimensional Projective Space

With the use of only the incidence axioms we prove and generalize Desargues’ two-triangle Theorem in three-dimensional projective space considering an arbitrary number of points on each one of the two distinct planes allowing corresponding points on the two planes to coincide and three points on any of the planes to be collinear. We provide three generalizations and we define the notions of a generalized line and a triangle-connected plane set of points.

Thomas Reid’s geometry of visibles and the parallel postulate

Thomas Reid (1710–1796) presented a two-dimensional geometry of the visual field in his Inquiry into the human mind (1764), whose axioms are different from those of Euclidean plane geometry. Reid’s ‘geometry of visibles’ is the same as the geometry of the surface of the sphere, described without reference to points and lines outside the surface itself. Interpreters of Reid seem to be divided in evaluating the significance of his geometry of visibles in the history of the discovery of non-Euclidean geometries. The question will be reexamined with particular attention given to his unpublished manuscripts. These include comments on Saccheri’s work and Reid’s repeated attempts to derive Euclid’s parallel postulate from the axioms of incidence.

Ordered Incidence geometry and the geometric foundations of convexity theory

An Ordered Incidence Geometry, that is a geometry with certain axioms of incidence and order, is proposed as a minimal setting for the fundamental convexity theorems, which usually appear in the context of a linear vector space, but require only incidence, order (and for separation, completeness), and none of the linear structure of a vector space.

Analoga der steinerschen konstruktion in einer absoluten geometrie

We consider an absolute geometry with the following base of axioms: Hilbert's plane axioms of incidence, order and congruence and a circle axiom. Thus no parallelism and not much continuity is involved. In this geometry the metric cannot be determined by Steiner's basic structure “fixed circle with centre”. In this work it will be proved that the following basic figures are suitable for such an absolute geometry in the sense that, after tracing any one of them, all constructions of second order can be done only with a ruler: 1) Two non-concentric circles, one of them with centre. 2) A unit-turner and a non-concentric circle without centre. 3) A circle with centreO and a line segmentA B with midpointM, the linesA B andO M being not orthogonal. 4) A circle with centre and two orthogonal lines, none of them passing through the centre. 5) A circle with centre and a distance-line (with their two branches).

A Characterization of Some Geometries of Chains

The geometries considered here are the Möbius plane M() (W. Benz [1]), the Laguerre plane L() (W. Benz and H. Mäurer [7]) and the Minkowski plane A() (W. Benz [5], G. Kaerlein [18]) over a field . All of them are geometries of an algebra with identity over a field. The characterization of the projective plane over a field by the proposition of Pappus first gave a close relation between algebraic and geometric structures. B. L. v. d. Waedern and L. J. Smid [28] presented a further example by characterizing the Möbius and Laguerre plane with incidence axioms and the "complete" proposition of Miquel.

A pascal theorem applied to Minkowski geometry

If on an oval in a projective plane a 4-point Pascal theorem, π, with fixed points U and V holds, then the oval is {(x,y) ¦xy=c} ∪ (O) ∪ (∞), with c ≠ O, in some Hall coordinatization. If for every 3 distinct points P, Q, R (not on UV; neither U nor V collinear with two of P, Q, R) there is through them a certain point set satisfying an extended version of π, then all these sets together with all lines not through U or V form the circles of a plane Minkowski (= pseudoeuclidean) geometry over a commutative field. π may be expressed in terms of Minkowski geometry. Together with incidence axioms derived from the protective incidence axioms, the Minkowski version of π characterizes the plane Minkowski geometry over a commutative field and is thus equivalent to Miquel's theorem.

Incidence axioms for affine geometry

In terms of incidence alone it is possible to define an affine plane, as Artin does [I], by calling lines parallel if they do not intersect, and basing the definition on the Euclidean axiom that there is a unique parallel to a line through a point not on the line. In higher dimensions we can define afine geometry by deleting the points and lines of a hyperplane from a projective geometry, using the axioms of Veblen and Young [4]. It is an easy exercise to show that the Artin approach and that of Veblen and Young agree in the definition of an afIine plane. Sut in higher dimensions it is not clear how an afline geometry can be defined directly so that it can be shown to arise from a projective geometry by deleting the points and lines of a hyperplane. This paper gives a set of axioms which have this property. We must define parallelism in such a way that nonintersecting lines in a plane are parallel. But the real surprise is that this is not enough. In Section 5 an example is given of a geometry in which every plane is an afhne plane, but the geometry as a whole is not an afline geometry. If we think of the embedding of an affine geometry into a projective geometry, lines are parallel if and only if they pass through the same projective point at infinity. In particular, the abstract property of parallelism between lines must be transitive, and three parallel lines need not lie in a plane. Thus the axiom of transitivity of parallelism (Axiom A4 in Section 2) is a three-dimensional axiom. The axioms are given in Section 2. Affine planes are defined in Section 3, and their main properties are developed there. The issue as to whether or not the plane is Desarguesian does not arise. In Section 4, the ideal (infinite) points and lines are defined and adjoined to the affine geometry. It is shown that the resulting geometry is a projective geometry. From the treatment here, it follows that if we have an incidence geometry, consisting of points and certain distinguished subsets of points which we call

Geometric structure with Hilbert's axioms of incidence and order

Geometric structure with Hilbert's axioms of incidence and order

A Problem in Projective Geometry

The theorem to be discussed in this note depends only on the plane incidence axioms (with Desargues), the Pappus axiom, and the property that a point and its harmonic conjugate are in general distinct. When algebra is applied to the geometry it may be taken over any commutative field of characteristic other than 2.

A theorem on parallels

For the theorem which we wish to prove we need Hilbert's axioms of incidence I 1-3 referring to plane geometry and the axioms of order [2, pp. 3-5]. Special emphasis will be placed on the second axiom of order, II 2: If A and C are two points of a straight line, there exists at least one point B of the line so situated that C lies between A and B. Now the main purpose of our theorem is to show the existence of parallels using continuity and not congruence as is usually the case in plane geometry. Therefore we have to state our axioms of continuity in a form not involving congruence, known as the postulate of Dedekind [1, p. 23 ]: For every partition of all the points of a segment into two nonvacuous sets, such that no point of either lies between two points of the other, there is a point of one set which lies between every other point of that set and every point of the other set. We prove the following

On the foundations of inversion geometry

Introduction. A 3-dimensional inversion geometry over an ordered field V in which every nonnegative number is a square may be defined as a partially ordered set II of objects called circles, spheres, and inversion space with the properties: (i) if p is any point, then there is an affine geometry whose points, lines, planes, and 3-space are, respectively, the points of II other than p, the containing p, the containing p, and the inversion space; (ii) the underlying field of this affine geometry is V; (iii) this affine geometry can be made a Euclidean geometry in such a way that the circles and spheres of the Euclidean geometry are, respectively, the of II not containing p and the of II not containing p. The purpose of this paper is to give axioms for II that will be sufficient to establish (i), (ii), and (iii). The only undefined relation is the ordering relation <, which means, geometrically, that all our axioms are incidence axioms. There does not seem to be any particular interest in finding alternative statements of (i), so (i) is simply assumed (1.4). Additional assumptions are added (2.11 and 2.12), and the remainder of the paper is devoted to proving that these axioms are sufficient for (ii) and (iii). The extension of this work to higher dimensions is straightforward, and we have concentrated on the 3-dimensional case for the sake of simplicity. The 2-dimensional case, however, is different in many ways('), and will be treated in a future paper. It is rather surprising that the literature contains so few investigations of the foundations of inversion geometry as an autonomous subject(2). Certainly much less is known about the postulates for inversion geometry than for other geometries. The present paper is an effort to remedy this deficiency. We wish to thank H. S. M. Coxeter, Tong Hing, and E. R. Lorch for their invaluable advice at various stages in the preparation of this manuscript. 1. The first set of postulates. In this section, we postulate that our set

On Desargues Theorem

The usual proofs of Desargues Theorem employ either metrical or analytical methods of projection from a point outside the plane; and if it is attempted to translate the analytical proof by the von Stuadt-Reye methods, the result is very long and there is trouble with coincidences. It is the object of this note to give a short geometrical proof which, in addition to the usual axioms of incidence and extension, uses only the assumption that a projectivity which leaves three points on a line unchanged also leaves all points on it unchanged. Degenerate cases are excluded as having no interest.

On the Theorem of Pappus

1. The axioms necessary for the construction of a proof of Pappus' Theorem, regarded as a theorem in the geometry of projective space of three dimensions, fall into three groups:I. Axioms of Incidence;II. Axioms of Order, giving the properties of the relation “between”, and establishing the order type of the projective line as cyclical and dense in itself;III. An Axiom of Continuity.It is customary in treatises on projective geometry to adopt in Group III the Axiom of Dedekind, which states that if the points of a segment are divided into two classes, L and R, which have each at least one member, and are such that no member of L lies between two points of R, nor vice versa, then there is a point of the segment, not an end-point, which is neither between two points of L nor between two points of R. This axiom, however, assumes considerably more than is necessary for the proof of the theorem.