Affine geometry

The affine geometry is a generalization of Euclidean geometry in which, although the Euclidean parallel axiom is true, but the distance and angle have no meaning. The term " affine geometry " is used for the mathematical sub-region and thus for the described " premises " of points and lines ( and derived therefrom, levels etc.). An affine geometry as a space is also referred to as affine space. It should be noted that any affine space, as it is characterized Linear Algebra, also meet the requirements of an affine geometry is sufficient, but not vice versa. The affine geometry generalizes the well-known concept from linear algebra. This article is the more general term, with which the synthetic geometry is concerned, therefore referred to throughout as " affine geometry ".

For the purposes of the Erlangen program of Felix Klein affine geometry can also be introduced as the epitome of the invariant under bijective affine geometric properties.

Definition

From an affine geometry is when a lot of points, a lot of straight lines, an incidence relation between and, and a parallelism relation is given, and the following axioms are satisfied:

Writing and speaking modes, basic properties

• Points are denoted by capital Latin letters.
• Lines are denoted by small Latin letters.
• Applies for and they say, A is incident with g, or A lies on g or g goes through A.
• Applies to so they say, g and h are parallel.
• The a pair after the fourth axiom clearly given parallels with occasionally as quoted.

Incidence of set theory

• The amount of points that are incident with a given line, called carrier set of lines, this amount is often used as noted. formalized:
• From the first two axioms follows that two straight lines coincide if and only if they are incident with the same points, that is, if their amounts are the same carrier. For this reason, it is often assumed equal in the recent literature assume that each is precisely the amount of inzidierenden with her points, ie. Then if and only if the following applies and the incidence relation can be completely replaced by the set-theoretic contain - relation.

Levels

• It follows from the third axiom that every straight line is parallel to itself, from the fourth follows then that straight lines that are parallel and have a point in common are identical. In other words, if two straight lines of different and parallel, then they are disjoint.
• Disjoint lines need not be generally parallel.
• The fifth axiom can also formulate equivalent using the fourth as:

• In particular, the point of the fifth axiom is uniquely determined.
• If there is now a triangle, three points that do not lie on the same straight line, then you can define with the fifth axiom a meaningful notion of a level that is determined by the triangle. One possible definition is as follows: A point is right then when the lines and cuts.

• From the fifth axiom, you can now ( with some technical effort and several case distinctions ) demonstrate that for straight lines that lie in, the following applies: If two straight lines of the plane disjoint, then they are parallel. To meet these "levels" all the axioms of an affine plane.

In summary:

• An affine geometry, in addition, the Reichhaltigkeitsaxiom

• If and only if also applies, that is, when there are not four points which do not lie on a common plane, the affine geometry is an affine plane.

Examples

• By vector spaces over a field generated affine spaces: The Euclidean space of intuition can be generated by a three-dimensional vector space over.
• Euclidean plane can be generated by a two-dimensional vector space.
• Trivial examples are: No point, no straight line ( -dimensional affine geometry)
• A single point and not a straight line ( zero-dimensional affine geometry)
• A straight line on which all points are (one-dimensional affine geometry)

• The smallest affine geometry, which includes a plane, the affinity level that can be generated by the two-dimensional vector space over the finite field. It consists of the points and the straight lines connecting lines consist in the right of the two specified points. Moreover applies. → See also the illustrations in affine plane.

Desargues and nichtdesarguessche geometries

All of vector spaces over a field, and even all in the same way produced by left vector spaces over a skew field affine geometries meet the large affine set of Desargues, they are affine spaces in the sense of linear algebra. For at least three-dimensional affine geometries the converse also holds: You can always describe by left vector spaces over a skew field. There are also flat affine ( " nichtdesarguessche " ) geometries ( → see affine plane) that do not satisfy the Desargues theorem. They can therefore not be generated by a vector space. Instead, you can give them as a coordinate space always assign a Ternärkörper.

Embedding problem and coordinate ranges

An affine space (in the sense of linear algebra ) is always together with his range of coordinates, a (skew - ) body and a - ( left ) vector space defined ( with the exception of the empty affine space, but to a subspace of a given space is considered a skew field ). In linear algebra, limited usually to vector spaces over commutative bodies, but the essential geometric facts (except the set of Pappus ) also apply more generally for left vector spaces over skew fields.

This is true for affine spaces:

Now apply to affine geometries:

• If the geometry contains a level, but does not coincide with her, then she is desarguesch and their incidence structure and its parallelism is a clear division ring and a unique dimension ( at least 3) for the " coordinate vector space " is given.

• If the geometry is a plane that satisfies the set of Desargues, the same is true with the dimension 2

It is in these cases, the concepts of affine geometry and affine space are equivalent. It takes in points 1 to 4 simply the concepts of linear algebra.

• A nichtdesarguesche level also determines a unique coordinate structure, a Ternärkörper, which, however, generally has much weaker properties than a skew field.

• For zero - and one-dimensional geometries that do not occur as subspaces of at least two-dimensional geometries, apparently a structural analysis is uninteresting: Through their incidence structure is given nothing more than the pure point set out.

A generalization of the concept of affine geometry, the concept of weak affine space. Each affine geometry is also a weak affine space. Some nichtdesarguessche affine planes are real subspaces of low-affinity spaces, although such levels can be embedded in broader affine geometries never.