Non-intersecting Lines Research Articles

A Crowded Set of Non-Intersecting Lines

A Crowded Set of Non-Intersecting Lines

A Crowded Set of Non-Intersecting Lines

A Crowded Set of Non-Intersecting Lines

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

O konstrukcích obecné plochy kubické, určené třemi mimoběžnými přímkami a 7 body, z nichž alespoň 3 jsou reálné

O konstrukcích obecné plochy kubické, určené třemi mimoběžnými přímkami a 7 body, z nichž alespoň 3 jsou reálné

On a group of order 25920 and the projective transformations of a cubic surface

The figure in three dimensions consisting of two sets of three non-intersecting lines, such that each line of one set intersects each line of the other set, will here be called a “net.” All nets are projectively equivalent to each other: and there are just 72 projective transformations that change one net into another.

Triads of ruled surfaces

Py, (yl, Y2, Y3, y4) ; Pz, (z1l Z2, Z3, Z4) ; Pa, (a,) a2, a3, a4) ; P#, (#l) 32, /33 (4), which in turn determine two non-intersecting lines ly, la#. As x varies these points trace four curves Cy, Cz, Ca, C,, while the lines ly, la# generate two ruled surfaces Ry, Ra#, on the first of which lie the curves Cy, Cz, and on the second, the curves Ca, Co. In this paper we propose to develop the projective differential theory of triads of ruled surfaces whose generators are in one-to-one correspondence. We will determine the defining system of differential equations, calculate certain of the invariants and covariants, and exhibit their geometric significance in a number of instances.

Extension of Some Definitions and Propositions in Euclid’s Book XI. And Remarks

Euclid does not define the inclination to one another, or the mutual perpendicularity of non-intersecting straight lines. (For brevity I shall omit the adjective straight before lines.)The following definition is justified by the 10th Proposition of Bk. XI. Def. : The inclination of two non-intersecting lines to one another, or the “ angle between them ” is the angle between two intersecting straight lines which are parallel to them respectively.

156. To Prove by Pascal's Theorem That the Straight Lines Meeting Three Non-Intersecting Straight Lines Generate a Conicoid, i.e. a Surface Every Plane Section of Which Is a Conic

156. To Prove by Pascal's Theorem That the Straight Lines Meeting Three Non-Intersecting Straight Lines Generate a Conicoid, i.e. a Surface Every Plane Section of Which Is a Conic

Několik analytických studií o plochách mimosměrek (zborcených). [I.

Několik analytických studií o plochách mimosměrek (zborcených). [I.

Několik analytických studií o plochách mimosměrek (zborcených). [III.

Několik analytických studií o plochách mimosměrek (zborcených). [III.

Několik analytických studií o plochách mimosměrek (zborcených). [IV.

Několik analytických studií o plochách mimosměrek (zborcených). [IV.

Několik analytických studií o plochách mimosměrek (zborcených). [II.

Několik analytických studií o plochách mimosměrek (zborcených). [II.

Několik analytických studií o plochách mimosměrek (zborcených). [V.

Několik analytických studií o plochách mimosměrek (zborcených). [V.