Diana: Created page with " == Definition == Let P be a set, which elements are called points, L is a collection of subsets of P, called lines and I is a relation between points and lines, called rela..."

2020-03-04T16:54:25Z

Created page with " == Definition == Let P be a set, which elements are called points, L is a collection of subsets of P, called lines and I is a relation between points and lines, called rela..."

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== Definition ==

Let P be a set, which elements are called points, L is a collection of subsets of P, called lines and I is a relation between points and lines, called relation of incidence.

The triple $Pi$=(P,L,I) is called '''a projective plane''', if
1. any pair of distinct points are incident with exactly one line;
2. any pair of distinct lines is incident exactly with one point;
3. there exists four points no three of which are incident with the same line.

Points which are incident with the same line are called collinear. \\
For any projective plane $\Pi$ there exists integer $q\geq 2$ such that
1. Any point (line) of projective plane $\Pi$ is incident exactly with $q+1$ lines (points).
2. A projective plane $\Pi$ has exactly q<sup>2+q+1$ points (lines).

This number q is called '''the dimension of projective plane''' and $\Pi$ is denoted by PG(2,q).

'''Colliniation of projective plane''' is an authomorphism of projective plane which preserve incidentness. $P\Gamma L(3,q)$ is the group of all colliniations of a projective plane of order q.

'''k- arc''' is a set of k point of a projective plane no three of which are collinear.

In a projective plane of order q the maximal size of k-arc is q+1, if q$is odd and q+2, if q is even. (q+1)-arc is called '''oval''', (q+2)-arc - '''hyperoval'''. Hyperovals exist only in projective planes of even dimension.

A line is called '''tangent''' to an oval if it meets the oval in precisely one point.

There is a unique tangent line to each point of oval in projective plane of even dimension. All tangent lines of an oval has intersection at one point and all lines through this point are tangent lines to the oval. The intersection point of all tangent lines to an oval is called '''nucleus'''.

In projective planes of even dimension every oval is contained in a unique hyperoval. This hyperoval is obtained by adding to the points of oval the nucleus. On the other hand, if we start with a hyperoval H and remove a
point N ∈H then we are left with an oval H\{N} which has nucleus N and completes to H. Thus, one hyperoval give rise to q+1 ovals.

Projective plane - Revision history

Diana: Created page with " == Definition == Let P be a set, which elements are called points, L is a collection of subsets of P, called lines and I is a relation between points and lines, called rela..."