# Difference between revisions of "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, , 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 . A plane consisting of a finite number of points, and thus of straight lines, is called finite .

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

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 .

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.

How to Cite This Entry:
Plane. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Plane&oldid=44355
This article was adapted from an original article by V.V. Afanas'evL.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article