Projective plane

A projective plane is a geometry points and lines comprehensive incidence structure that is characterized essentially by two requirements, namely that any two lines a ( unique) intersection point and two points have a (unique ) connecting line.

• 4.1 lock sets
• 4.2 Koordinatisierung
• 4.3 collineations

Definition

An incidence structure is called projective plane, if the following holds:

• There are two distinct points is exactly one line incident with both.
• There is exactly one point incident with both For any two distinct lines.
• There is a complete quadrilateral, that is, four points, no three of which are incident with the same straight line.

Examples

• If one conceives as points in the three-dimensional vector spaces over the real numbers or the complex numbers, the two-dimensional subspaces as straight lines and the one-dimensional subspaces, one obtains models of a projective plane. The incidence relation is the usual inclusion. Those levels with the similar levels obtained on the quaternions or octonions are also referred to as a traditional levels. Instead of the real or complex numbers, one can take an arbitrary field K, even a skew field (the most famous is composed of the quaternions ). Such projective planes over skew fields meet all the set of Desargues and are therefore referred to as desarguesian projective planes.
• The smallest possible finite projective plane ( minimal model ) consists of seven lines and seven points (see figure). In this case, K is the body made only from 0 and 1 and in the 1 1 = 0, so the residue field.
• There are also nichtdesarguessche projective planes. They can be koordinatisiert by ( finite or infinite ) Ternärkörper in a similar manner as the Desargues by skew field. → See also classification of projective planes.

Comments

Principle of duality

One can show that there are always four lines in a projective plane of which no three go through the same point. From this and from the symmetric formulation of the first two axioms can be seen that we obtain point and line again a projective plane by swapping the names. The points and lines of the form straight lines and points to the dual plane.

Connection with affine planes

Taking in an affine plane for each family of parallel straight lines a further point at infinity added to, which is incident with exactly the lines of his flock, and it extended to the improper line that contains exactly these points, we obtain a projective plane, the projective completion of. Conversely, we obtain an affine share a projective plane by removing any arbitrary straight line with all their points. It should be noted:

The projective planes in which all slotted cutouts are yet another isomorphic affine planes are exactly the Moufangebenen.

Finite levels

As the minimal model described above shows, projective planes can be finite, ie, contain only a finite number of points and lines. Contains a straight line n 1 points, so all lines contain n 1 points, go through each point n 1 lines and in total there are n ² n 1 lines and n ² n 1 points. n is in this case the order of level. A finite projective plane of order can be combinatorially as a symmetric 2 - block plan conceived. The smallest possible order of a finite projective plane is two. For each order, which is a prime power, can construct a finite projective plane. Whether there is such a plane whose order is not a prime power, is an unsolved problem. Part Results: The non-existence of a projective plane of order 10 was demonstrated with a large computing. The set of states Bruck- Ryser- Chowla: If n = 4k 1 or 4k 2 order of a projective plane, then n is the sum of two whole squares. Then there is no projective planes of systems 6, 14, 21, 22, 30, 33, 38, 42, 46, .... Whether those of the orders 12, 15, 18, ​​20, 24, 28, ... there is unknown.

Can be completely described by a set of only natural numbers A particular class of finite projective planes of order n: the levels derived from a difference set. It is known that every finite level desarguesian this class belongs and it is believed that each level of this class is desarguessch.

The real projective plane as a quotient set of a sphere

In some respects, particularly in terms of topology, can be a real projective plane understood as that which is obtained when going on a sphere ( surface of a sphere in 3-dimensional space ) each antipodes, ie points of the sphere, the ones on ends of a diameter are, " equates ". More specifically, this means that one takes as points of the projective plane each pairs of antipodes and as straight lines the same great circles, ie circles, the intersection of the sphere with a plane passing through the center of spheres ordinary level. Thus, the real projective plane is topologically the quotient topology of the sphere.

The sphere itself is an orientable surface, the resulting through this process of forming the quotient projective plane it is no longer, as the antipodal map as a reflection about the center is no rotation. The real projective plane may be represented as a surface in three-dimensional space image. Examples include the Boy's surface, the cross-cap and the Roman surface. Just as with the also non- orientable Klein bottle can not be an embedding of the projective plane in three-dimensional space without self-intersection.

Classification

Lock sets

An obvious a classification of projective planes is purely due to the concept of incidence. This is done by determining whether certain geometric sentences of the form " if a particular configuration of incidences is present, then a further incidence applies " in a plane apply. Examples of such lock sets are known from the real plane ( and valid there ) sets of Desargues and Pappus (sometimes called set of Pappus -Pascal ). Planes in which the said rates shall be called the Desargues planes or Pappossche levels. One level in the small set of projective Desargues is universal, ie Moufangebene. Each level is pappossche desarguesch and each level desarguesche a Moufangebene.

Koordinatisierung

Methods for utilizing the algebra another common geometry in the process is the introduction of co-ordinates. These establish a connection between the geometric structure of the plane and the algebraic one underlying coordinate space. In any projective plane coordinates can be introduced: this is a projective point base is selected in the plane is a straight line to the line at infinity determined (→ see projective coordinate system). Then on the affine plane which is formed by cutting out the line at infinity, are constructed with a Ternärverknüpfung, which can be described purely geometric coordinates as a amount Ternärkörper. The computation rules in a body are in the corresponding coordinate range, the Ternärkörper, generally not.

There is a direct relationship between the geometric structure of the plane and the algebraic the coordinate region which characterizes the level in a certain way. The Moufangebenen are, for example, exactly the projective planes whose coordinate space is an alternative body, the Desargues planes are exactly those that have a skew field as a coordinate space. If the coordinate space a commutative field, then the level is pappossch. In this case, one usually uses homogeneous coordinates ( → see the main article Homogeneous coordinates). From the theorem of Wedderburn shows that finite desarguesian levels are always pappossch. Ruth Moufang able to prove that even every finite Moufangebene is pappossch.

Collineations

The straight loyal bijections are the structure preserving mappings (or isomorphisms ) between projective planes. Such a bijection is the points from the points and the straight line on the straight line in such a way that the incidence is maintained. The collineations that are just loyal bijections a projective plane onto itself form a group, called the collineation group of the plane. Examples of collineations, operating in the slotted projective plane, so as affine collineations are translations or rotations and general affinities.

In the projective plane, even the group of projectivities is a subgroup of the collineation. This sub-group is defined as a product in geometry of the synthetic part of the amount of the Perspektivitäten collineation, examination of operations of certain subsets of the collineation on the plane represents a further possibility of classification dar.