# Level (math)

The plane is a basic concept of geometry . In general, it is an infinitely extended flat two-dimensional object.

• Here, unlimited, extended and flat means that for every two points, a straight line running through this also lies completely in the plane.
• Two-dimensional means that - apart from the straight lines it contains - no real subspace also has this property.

More concretely, a plane is used to describe different objects , depending on the sub-area of mathematics .

## Level as an independent object

smallest projective plane (seven points, seven straight lines)
smallest affine plane (four points, six straight lines)

### The classic concept of plane according to Euclid

In classical geometry, for example in the sense of Euclid's elements , the (Euclidean) plane - usually referred to in this context with the specific article - forms the framework for geometrical investigations, e.g. for constructions with compasses and ruler . You can imagine it as an abstraction of the drawing plane (paper) as infinitely extended and infinitely flat, just as the straight line is an abstraction of the drawn line (pencil line) imagined as infinitely thin and infinitely long. The Euclidean geometry is nowadays by Hilbert's axioms described.

Since Descartes provided the Euclidean plane 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 Hilbert's axioms of the plane. This real vector space is therefore also called a plane. ${\ displaystyle \ mathbb {R} ^ {2}}$ ${\ displaystyle \ mathbb {R} ^ {2}}$

### The projective plane

If you supplement Euclid's affine plane with an infinitely distant straight line and infinitely distant points lying on it, you get a projective plane.

The projective level can also be described algebraically , namely as the set of all one-dimensional subspaces in the . The straight lines running through the origin are therefore understood as points of the projective plane. The straight lines of the projective plane are then exactly the two-dimensional sub - vector spaces of , that is, the "conventional" planes running through the origin. ${\ displaystyle \ mathbb {R} ^ {3}}$${\ displaystyle \ mathbb {R} ^ {3}}$

### Generalizations

If one weakens Hilbert's system of axioms, even finite structures are possible, which are also referred to as the affine plane or the projective plane. The figure on the right shows a finite projective plane with seven points and seven straight lines. By removing any straight line and the points lying on it, you get a finite affine plane with four points and six straight lines.

In a generalization of the Cartesian model of the Euclidean plane, the two-dimensional vector space is also referred to as an affine plane for any bodies ; correspondingly for the projective plane. Note: If the body of the complex numbers, which are illustrated by the Gaussian plane of numbers, is already (real) two-dimensional, but is referred to as a complex straight line. The plane is really four-dimensional, but only a two-dimensional complex vector space. The body can also be a finite body . In the case one obtains the smallest finite affine plane described above with four points or the projective plane with seven points. ${\ displaystyle K}$${\ displaystyle K ^ {2}}$${\ displaystyle K}$${\ displaystyle \ mathbb {C}}$${\ displaystyle \ mathbb {C}}$${\ displaystyle \ mathbb {C} ^ {2}}$${\ displaystyle K}$${\ displaystyle K = \ mathbb {F} _ {2}}$

A surface in the sense of topology is the plane (also the projective one) only in the case ; in the case it is a complex surface . ${\ displaystyle K = \ mathbb {R}}$${\ displaystyle K = \ mathbb {C}}$

## Level as a subspace

Two intersecting levels

If one considers higher-dimensional geometric spaces, then each subspace that is isomorphic to a plane in the above sense is called a plane. In a three-dimensional Euclidean space , a plane is defined by

• three non-collinear points
• a straight line and a point not lying on it
• two intersecting straight lines or
• two really parallel straight lines

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

Two planes are either parallel, intersect in a straight line or are identical. They cannot be skewed to one another in (three-dimensional) space.

• In the first case, every straight line perpendicular to the first plane is also perpendicular to the second. The length of the line that the planes delimit such a straight line is called the distance between the planes.
• In the second case one considers a plane perpendicular to the line of intersection. The first two levels intersect with this in two straight lines. The angle between these straight lines is called the angle between the two planes.

Every two-dimensional subspace of the coordinate space (or ) forms a plane of origin , i.e. a plane that contains the zero point of the space. Affine two-dimensional subspaces are parallel shifted planes that do not contain the zero point. ${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle K ^ {n}}$

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

## Plane equations

Representation of a plane in parameter form

Planes in three-dimensional space can be described in various ways by plane equations. A plane then consists of those points in a Cartesian coordinate system whose coordinates satisfy the plane equation. A distinction is made between explicit forms of plane equations, in which every 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 parametric form and the three-point form , the implicit forms include the normal form , the Hessian normal form , the coordinate form and the intercept form .

When describing planes in higher-dimensional spaces, the parametric form and the three-point form retain their representation, with only -component instead of three-component vectors being used. Due to the implicit forms, however, a level is no longer described in higher-dimensional spaces, but a hyperplane of the dimension . However, each plane can be represented as an intersection of hyperplanes with linearly independent normal vectors and must therefore satisfy as many coordinate equations at the same time. ${\ displaystyle n}$${\ displaystyle n-1}$${\ displaystyle n-2}$