Plane (geometry)

The plane is a basic concept of geometry. In general, there is a vast unlimited flat two-dimensional object.

  • This means unlimited extended and flat, that for every two points a straight line lies completely through this running in the plane.
  • Two-dimensional means that - apart from straights contained - no proper subspace also has this property.

More specifically, we denote by plane, depending on the branch of mathematics, however, quite different objects.

Level as an independent object

The classic plane phrase Euclid

In classical geometry in the sense of Euclid's Elements, the ( Euclidean ) plane forms - in this context usually referred to with the definite article - the frame geometric investigations, eg for constructions with ruler and compass. One can imagine extended as an abstraction of the plane (paper) as infinite and infinitely flat, as the straight one presented as infinitely thin and infinitely long abstraction of the drawn stroke (pencil line). Euclidean geometry is now described by Hilbert's system of axioms of Euclidean geometry.

Since Descartes, the Euclidean plane is equipped with coordinates, one can identify the Euclidean plane with the set of all ordered pairs of real numbers. Or the other way around: forms a model for the Hilbert 's axioms of the plane. This real vector space is therefore also referred to as level.

The projective plane

Supplementing Euclid affine plane about a line at infinity and lying on their points at infinity, we obtain a projective plane.

The projective plane can be described algebraically, namely as the set of all one-dimensional subspaces in. So You pick up the line passing through the origin straight on as points of the projective plane. The straight lines of the projective plane are then exactly the two-dimensional subspaces of, ie the line passing through the origin of " conventional " levels.


Weakens to Hilbert's axiom system from, so even finite structures are possible, which are also called affine or projective plane. The figure at right shows a finite projective plane with seven points and seven lines. By removing any straight line and the points lying on it to get a finite affine plane with four points and six lines.

As a generalization of the Cartesian model of Euclidean plane of the two-dimensional vector space is called the affine plane and for any body; mutandis to the projective plane. Note: If the field of complex numbers, which are indeed illustrated by the Gaussian number plane, so ( real) is already in two dimensions, but is referred to as a complex line. The level is real four-dimensional, but only a two-dimensional complex vector space. The body can also be a finite field. In case one obtains the above-described smallest finite affine plane with four points and the projective plane with seven points.

An area in the sense of the topology is the plane (also projective ) in the case; in case it is a complex area.

Level as a subspace

Considering higher-dimensional geometrical space, each partial space as referred to, which is isomorphic to a level as defined above, as a plane. In a three-dimensional Euclidean space, a level set by this

  • Three non- collinear points
  • A line and a point not lying on it
  • Two intersecting straight lines or
  • Two real parallel lines

If two straight lines to one another skewed, so they do not lie, however, in a common plane. Instead, there are then two parallel planes, each of which each contains one of the straights.

Two planes are either parallel, intersect in a straight line or are identical. You can lie askew to each other in the ( three-dimensional ) space so do not. In the first case, each of the first plane perpendicular line and perpendicular to the second. The length of the track, the limit levels in such a line is referred to as the distance between the planes. In the second case we consider a perpendicular plane to the sectional line. With this, the first two planes intersect in two straight. The angle between the straight line is referred to as the angle between the two planes.

Each two-dimensional subspace of the space coordinates (respectively) forms an original plane, ie a plane containing the origin of the area. Two-dimensional affine subspace in parallel displaced planes which do not include the zero point.

Not every falling under the concept of level mathematical object can be interpreted as subspace of a corresponding higher-dimensional space. For example, the Moulton plane is an affine plane in which the set of Desargues does not apply, while in each three-dimensional affine space - always applies - and thus contained in each level.

Plane equations

Levels in the three-dimensional space can be described in different ways through plane equations. A level then consists of those points in a Cartesian coordinate system whose coordinates satisfy the plane equation. A distinction is made explicit forms of plane equations, where each point of the plane is identified directly, and implicit forms in which the points of the plane are indirectly characterized by a condition. The explicit forms include the parameter form and the three -point form, to the implicit forms of the normal form, the Hessian normal form, the coordinate form and intercept form.

In the description of levels in higher-dimensional spaces, the parameter form and the three -point form retain their representation, whereby only expected -component instead of three-component vectors. By the implicit forms, however, no level will be described more in higher dimensional spaces, but a hyperplane of dimension. However, each level can be represented as the intersection of hyperplanes with linearly independent normal vectors, and therefore must comply with the same number of coordinates equations simultaneously.