View source for 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, [[#References|[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|Desargues geometry]]; [[Pascal 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|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 [[#References|[1]]]. A plane consisting of a finite number of points, and thus of straight lines, is called finite [[#References|[7]]].

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

In the analytic geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p0727901.png" />, a plane is a concept derived from the concepts of a  "vector"  and a  "point" . By the plane passing through a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p0727902.png" /> and through vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p0727903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p0727904.png" /> one understands the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p0727905.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p0727906.png" /></td> </tr></table>

In a rectangular coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p0727907.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p0727908.png" />, a plane is specified by a linear equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p0727909.png" /></td> </tr></table>

the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279010.png" /> of which define the coordinates of the normal vector for this plane. In an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279011.png" />-dimensional space, planes of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279012.png" /> are described by systems of linear equations [[#References|[5]]].

The mutual disposition of planes in various <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279013.png" />-dimensional spaces is determined by the corresponding incidence axioms, as is the incidence property for planes and straight lines.

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



====Comments====
For instance, for a [[Projective plane|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|Affine geometry]]) is a point-line incidence structure such that:

iv) for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279014.png" /> and line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279015.png" /> not incident with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279016.png" /> there exists a unique line through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279017.png" /> with no point in common with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279018.png" />;

v) there are three non-collinear points;

vi) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279019.png" /> ii).

For each affine plane there is an associated [[Projective plane|projective plane]], obtained by introducing one extra ideal point for each class of parallel (i.e. non-intersecting) lines and one extra line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279020.png" /> consisting of the ideal points. Conversely, a projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279021.png" /> with one (distinguished) line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279022.png" /> excluded is an affine plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279023.png" />.

A collineation of an affine or projective plane is a permutation (cf. [[Permutation of a set|Permutation of a set]]) of its points which takes lines into lines. A dilatation of the affine plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279024.png" /> is a collineation of the associated projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279025.png" /> which is the identity on the ideal line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279026.png" />. The affine plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279027.png" /> is a translation plane if all its dilations are transitive (i.e. for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279028.png" /> there is a dilatation taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279029.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072790/p07279030.png" />).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,   "Introduction to geometry" , Wiley  (1969)  pp. 178</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,   "Projective geometry" , Univ. Toronto Press  (1974)</TD></TR></table>