Plane

One of the basic concepts in geometry; it is usually indirectly defined in terms of the geometrical axioms. A plane may be regarded as a combination of two disjoint sets: A set of points and a set of straight lines, with a symmetric incidence relation between point and line. In accordance with the requirements satisfied by the incidence relation, which are described by certain axioms, one may distinguish projective, affine, hyperbolic, elliptic, and other planes.

Planes may be classified in terms of collineation groups (see, for example, [7], Chapt. 3, where the Lenz–Barlotti classification is given for projective and affine planes) or from the realization in the plane of various configurations (see, for example, Desargues geometry; Pascal geometry).

A plane is called metrical if the incidence relation is accompanied by a definition of distance between any pair of points. For example, in the Hilbert system of axioms of Euclidean geometry, distance is introduced on the basis of congruence and continuity axioms, and the plane in that case is called continuous [1]. A plane consisting of a finite number of points, and thus of straight lines, is called finite [7].

One way for studying a plane is to introduce coordinates and a ternary operation, which is then examined [7], [8].

In the analytic geometry of $E^3$, a plane is a concept derived from the concepts of a "vector" and a "point" . By the plane passing through a point $A\in E^3$ and through vectors $\mathbf{m}$ and $\mathbf{n}$ one understands the set of points $M$ such that

$\overline{AM} = \mu \mathbf{m}+ \nu \mathbf{n}$, where $\mu,\nu\in \mathbb{R}$

In a rectangular coordinate system $(x,y,z)$ in $E^3$, a plane is specified by a linear equation

$Ax+By+Cz+D=0$

The coefficients $A, B, C$ of which define the coordinates of the normal vector for this plane. In an $m$-dimensional space, planes of dimension $n$ are described by systems of linear equations [5].

The mutual disposition of planes in various $m$-dimensional spaces is determined by the corresponding incidence axioms, as is the incidence property for planes and straight lines.

References

[1]	D. Hilbert, "Grundlagen der Geometrie" , Teubner, reprint (1962)
[2]	N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)
[3]	, On the foundations of geometry. A collection of classical papers on Lobachevskii geometry , Moscow (1956) (In Russian) (Collection of translations)
[4]	F. Bachmann, "Aufbau der Geometrie aus dem Spiegelungbegriff" , Springer (1973)
[5]	A. Doneddu, "Géométrie euclidienne plane" , Dunod (1965)
[6]	B.A. Rozenfel'd, "Multi-dimensional spaces" , Moscow (1966) (In Russian)
[7]	R. Dembowski, "Finite geometries" , Springer (1968) pp. 254
[8]	G. Pickert, "Projective Ebenen" , Springer, reprint (1975)

Comments

For instance, for a projective plane the axioms are:

i) for each two lines there is a unique point incident with both of them;

ii) for each two points there is a unique line passing through them;

iii) there are four points no three of which are collinear.

This last axiom rules out special cases like the geometry of three lines intersecting in these points.

An affine plane (cf. also Affine geometry) is a point-line incidence structure such that:

iv) for every point $ p $ and line $ L $ not incident with $ p $ there exists a unique line through $ p $ with no point in common with $ L $;

v) there are three non-collinear points;

vi) $ = $ ii).

For each affine plane there is an associated projective plane, obtained by introducing one extra ideal point for each class of parallel (i.e. non-intersecting) lines and one extra line $ W $ consisting of the ideal points. Conversely, a projective plane $ P $ with one (distinguished) line $ W $ excluded is an affine plane $ A = P ^{W} = P \setminus W $.

A collineation of an affine or projective plane is a permutation (cf. Permutation of a set) of its points which takes lines into lines. A dilatation of the affine plane $ A $ is a collineation of the associated projective plane $ P $ which is the identity on the ideal line $ W = P \setminus A $. The affine plane $ A $ is a translation plane if all its dilations are transitive (i.e. for all $ p,\ q \in A $ there is a dilatation taking $ p $ to $ q $).

References

[a1]	H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969) pp. 178
[a2]	H.S.M. Coxeter, "Projective geometry" , Univ. Toronto Press (1974)