Plan refines incidental

In an axiomatic approach of the geometry, a plane refines incidental is the data of right points and with a relation of membership of the points to the right-hand sides, called incidence, checking the axioms of incidence :

Two points has and B is incidental on a single line (noted (AB)), and any line has at least two distinct points;
There exist at least three nonincidental points on the same line;
For any line D and any point has not incident with D , there exists a single incidental line $d'$ with has such as any point is not incidental with the two lines $d$ and $d'$ .

Parallelism

In this approach, a line D is known as parallel to $d'$ if these two lines are confused (equal) or although there does not exist any incidental point with the two lines. The lines $d$ and $d'$ are intersected in has if the point has is incidental with $d$ and with $d'$ . The incidental line unicity at two distinct points implies that two nonparallel lines are intersected in a single point. One has the following dichotomy:

Or two lines its parallels;
Or they are intersected in a single point.

Two parallel straight lines which are intersected are necessarily confused. In this case, they have at least two incidental points in commun runs.

The third axiom is reformulated by the existence and the unicity of a parallel on a given line passing by a given point.

Parallelism is a relation of equivalence:

By definition, any line is parallel to itself;
If $d$ is parallel to $d'$ , then it is easy to note that $d'$ is parallel to $d$ ;
transitivity is shown by a reasoning by the absurdity.

Let us suppose data three lines

d

d'

and

d

such as $d$ is parallel to $d'$ and that $d'$ is parallel to $d$ , but that

d

and

d

are not parallel. The three lines can be supposed two to two not confused. In this case, being not parallels, the lines $d$ and $d$ are intersected in a single point has . As parallelism is a symmetrical relation, each one of the right-hand sides

d

and

d

is parallel to $d$ . By unicity of the parallel on a line passing by a point, these lines are confused, which is contrary with the assumption.

Smaller plan refines incidental

If there exists, a plan refines incidental has at least four not aligned distinct items three to three.

Indeed, to start, there exists by the second axiom at least three distinct points has , B , C not incidents on the same line. In particular, C is not incidental with the right-hand side $(AB)$ . By the third axiom, there exists an incidental line $d$ with C parallel with $(AB)$ . Again by the first axiom, there exists a point D distinct from C which is incidental with $d$ . As D and $(AB)$ parallel and is not confused, they are not intersected. The point D is necessarily distinct from has and of B , and better still is not either incidental with $(AB)$ . It is followed from there directly that the points $A, B, C, D$ are two to two distinct and three to three not aligned.

The smallest plan refines incidental $P_0$ consists of:

Four distinct points $A$ , $B$ , $C$ and $D$ ;
And six lines, one and only one incidental with two unspecified of these four points.

The data of four items two to two distinct and three to three not aligned in an incidental infinite plan P provides a copy of

P_0

Notice on the models

The theory of the plans closely connected incidental is finiment axiomatisable. She asks for the use of at least:

a symbol of arite 1 relation allowing to distinguish the points and the lines;
a symbol of arite 2 relation allowing to define the incidence.

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