Linear Geometry Definition

A line may not have more than one point in common. 1.

In an intuitive way, this can be visualised as two rectilinear rules that never cut each other more than once. Finite linear space can be seen as a generalisation of projected and affine levels and more generally of 2-(v,k,1){\displaystyle 2-(v,k,1)} cluster design, wherein the demand that each cluster contains the same number of points is abandoned and the substantial textural property is that 2 points coincide with exactly 1 line.

In 1964 Libois invented the concept of linear space, although many results on linear space are much older. A = (P, I, G) is an incident pattern in which the element of points and the element of line are named points. If the following three maxims apply, a linear room is L:

L1) two points are connected by exactly one line. _GO ( 2 ) Each line applies to at least two points. L contains at least two rows. A few writers fall (L3) when they define linear rooms. The linear space corresponding to (L3) is not regarded as ordinary in such a case, and those which are not are regarded as ordinary.

With its dots and contours, the ordinary level of Euclidus forms a linear sphere; moreover, all affined and projected spheres are also linear spheres. All non-trivial linear dimensions of five points are shown in the following chart. Since any two points always coincide with a line, those that coincide with only two points are not conventionallyrawn.

The first figure does not show the ten line segments that connect the ten point couples. The second figure does not draw seven line segments that connect seven sets of dots. In the De Bruijn-Erd?s theory (geometry of incidence) it is shown that in every linear final room S=(P,B,I){\displaystyle S=( ({\mathcal {P}},{\mathcal {B}}, Shult, Ernest E. (2011), points and line, Universitext, Springer, doi:10.1007-3-642-15627-4, ISBN 978-3-642-15626-7 .

Finite linear space theories.