Affine plane (incidence geometry)

An affine plane is in synthetic geometry, points and lines comprehensive incidence structure that is characterized essentially by two requirements, namely that any two points have a (unique ) connecting line and that there is a clear parallel lines. In linear algebra and analytic geometry, a two - dimensional affine space is called the affine plane. The this article describes the concept of synthetic geometry generalizes this well-known concept from linear algebra.

An affine plane that contains only a finite number of points is referred to as a finite affine plane and examined as such in finite geometry. Especially for these levels of concept system level is important: It is defined as the number of points on a straight line and thus each level.

Every affine plane can be extended by introducing improper points and one of these existing inauthentic straight to a projective plane. Conversely, arises from a projective plane by removing a line with their points of an affine plane. → See also projective coordinate system.

Every affine plane can koordinatisiert by assigning a coordinate space and are algebraisiert by additional links that result from the geometric properties of the plane in this coordinate range. An affine plane in the sense of linear algebra, which is an affine space, the vector space of parallel shifts is a two - dimensional vector space over a field if and only if the coordinate space by the geometrical structure isomorphic results, to this very body is. This description of the affine plane by means of a coordinate space in which the body is generalized algebraic notion, and an overview of the structures that arise in validity major lock sets, can be found in the main article Ternärkörper.

On the other hand, one can examine the group of parallel shifts in an affine plane, which leads to another algebraization in which the term parallel shift, which can be described in linear algebra by a vector, leads to the concept of translation. This approach, which complements the coordinate- based access is described in the main article affine translation plane.

Definitions

An incidence structure consisting of a point space, a straight space and an incidence relation between these is an affine plane if and only if the following axioms hold:

Formalized, the three axioms record as:

Parallelism

The relation ( parallelism) between the straight line is defined by:

The uniquely determined by the second axiom straight line passing through a given point, is referred to as the parallel to and through as quoted.

This relation is an equivalence relation. The equivalence class of parallel to a straight line is called a parallel class as well as the direction of.

Ways of speaking

• The uniquely determined by the first axiom Just lie on the two different points is called a straight line connecting the points as, sometimes called quoted.
• The parallel class of a line is as listed.
• The referred to by a straight line and any point uniquely determined as the line is parallel with and as quoted.

The conventional point of view, in which the set of points and the set of lines were taken as first independent sets is based on more often in the recent mathematical literature. In this connection, then the set of points which lie on a straight line, referred to as set of points of the straight lines, and often as noted.

As a completely straight line but is defined by the incidence ratio, it is often identified by the set of points, so that the relationship is not necessary. The axioms are then described as properties of the set of lines, which is a subset of the power set of the set of points, the role of the incidence relation then takes the element relation: ( if and only if it is ).

Order of the affine plane

The order of an affinity level is defined as the thickness of the set of points on a straight line. The term is independent of the line, because all the lines of an affine plane ( as point sets ) are equally powerful, as two distinct lines can always be represented by a bijective projection parallel to each other. The following applies:

Every affine plane can be achieved by projective complete, that is, by adding a " line at infinity " along with their assign points ( as a remote elements of the affine plane ), a unique up to isomorphism projective plane. Every projective plane may be generated. It transfers the concept of order on the projective completion: The projective plane has order an arbitrary affine plane as its projective completion they can be constructed. This affine planes need not be isomorphic, but they always have the same order. If this order is equal to the finite number, then has the projective plane points and just as many straight lines on each line are exactly points and go through each point exactly straight.

Finite levels and open questions

• All currently known finite affine planes have a prime power as order and for any prime power exist affine planes with this order (as of 2013). What numbers occur is affine planes as orders an unsolved problem. From the set of Bruck and Ryser, a non- existence theorem for planes results with certain orders: For example, the numbers 6, 14, 21, 22, 30, 33, 38, 42, ... not orders affine planes. The ordinal 10 could be excluded by massive use of computers. 12 is the smallest number for which the existence question is unresolved.
• Is every affine plane of prime order desarguessch? This is an unsolved problem.
• If the order of every affine plane is a prime power? This question has not yet been clarified.

→ In general, the investigation of finite projective planes to their conclusion, the finite projective planes concentrated. An overview of the relationships between affine planes and their projective completion are the products Ternärkörper. Examples of structure and statements about nichtdesarguessche projective planes can be found in the article classification of projective planes.

Examples

• The two-dimensional vector space over the real numbers, with the proviso includes all one-dimensional affine subspaces, and the incidence relation is given by the relation Included.
• Similarly, the two-dimensional vector space over an arbitrary field (or: skew field ). Every affine plane in which the theorem of Desargues is isomorphic to an affine plane over a skew field. Applies in this plane plus the set of Pappus (also " set of Pappus -Pascal " ) as the division ring is a field ( with commutative multiplication).

Of particular interest is the nichtdesarguesschen levels have been found in which the set of Desargues does not apply. In them, you have coordinates from Ternärkörpern introduced, especially from quasi bodies ( also called Veblen - Wedderburn systems, with non- associative multiplication ) and Fast bodies ( in which is true of the two distributive only one).

• In case the result is the smallest affine plane. It consists of four points.

• There are affine planes with a finite number, say n points on a (and then every ) line. They are called n- th order or of the order n for every prime power q, there are affine planes of order q. Whether there are affine planes whose order is not a prime power, is an unsolved problem. A part of result is given by the set of Bruck and Ryser.

This states the following: Lets n when divided by 4 remainder 1 or 2 and n is order of an affine plane, then n is the sum of two squares of natural numbers. Examples: 6 is not an affine plane of order. 10 is not excluded by the theorem.

With great use of computers, however, the non-existence of an affine plane of order 10 was shown. Unresolved the question of existence is, for example, for orders 12, 15, 18, 20, 26, 34, 45, ..., and excluded the existence of n = 14, 21, 22, 30, 33, 38, 42, 46, ....

• The figures below show the minimal model of an affine plane ( left) and its projective extension, the minimal model of a projective plane.

Generalizations

• The affine plane is the two-dimensional special case of an affine geometry.
• Finite affine planes are among the networks. An affine plane of order n is a power.
• More generally include the finite affine planes as all nets to block plans and thus to the finite incidence structures. An affine plane of order n is a block diagram. As incidence structure is a finite affine plane is of type.