Affine geometry

The affine geometry is a generalization of Euclidean geometry , although the Euclidean in the parallel axiom is true, but have distance and angle of no importance. The term “affine geometry” is used for the mathematical sub-area and for the “spaces” described by it consisting of points and straight lines (and, derived from them, planes etc.). An affine geometry as space is also referred to as affine space . It should be noted that every affine space , as it is characterized by linear algebra , also meets the requirements of an affine geometry , but not vice versa. Affine geometry generalizes the more well-known term from linear algebra. Throughout this article, the more general term synthetic geometry is used to refer to as “affine geometry”.

In the sense of Felix Klein's Erlangen program , affine geometry can also be introduced as the epitome of the geometric properties invariant under bijective affine mappings .

definition

Of an affine geometry is when a lot of points , a lot of straight lines , one incidence relation between and as well as a parallel relation on is given and the following axioms are satisfied: ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle {\mathfrak {G}}}$ ${\displaystyle \mathrm {I} }$ ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle {\mathfrak {G}}}$ ${\displaystyle \|}$ ${\displaystyle {\mathfrak {G}}}$

1. Exactly one straight line (i.e. and ) passes through two different points , the connecting straight line (also written). ${\displaystyle A,\,B}$ ${\displaystyle g}$ ${\displaystyle AIg}$ ${\displaystyle BIg}$ ${\displaystyle g=AB}$ ${\displaystyle g=A\lor B}$
2. There are at least two points on every straight line.
3. The parallelism relation is an equivalence relation ${\displaystyle \parallel }$
4. Exactly one straight line passes through each point and is parallel to a given straight line.
5. If there is a triangle (three points not lying on a straight line) and two points and such that the straight line is parallel to the straight line , there is a point such that it is also parallel to and parallel to the line. ${\displaystyle ABC}$ ${\displaystyle A'}$ ${\displaystyle B'}$ ${\displaystyle AB}$ ${\displaystyle A'B'}$ ${\displaystyle C'}$ ${\displaystyle AC}$ ${\displaystyle A'C'}$ ${\displaystyle BC}$ ${\displaystyle B'C'}$

Spelling and speaking styles, basic characteristics

• Points are denoted with capital Latin letters. ${\displaystyle A,B,C,...\in {\mathfrak {P}}}$
• Straight lines are denoted with small Latin letters. ${\displaystyle a,b,c,...\in {\mathfrak {G}}}$
• Applies to, and so one says, A is incised with g, or A lies on g, or g passes through A. ${\displaystyle A\in {\mathfrak {P}}}$ ${\displaystyle g\in {\mathfrak {G}}\ A\mathrm {I} g}$
• This is true for so one says that g and h are parallel. ${\displaystyle g,h\in {\mathfrak {G}}\ g\|h}$
• The parallel to through , which is clearly given to a pair according to the fourth axiom , is occasionally noted as. ${\displaystyle (A,g)\in {\mathfrak {P}}\times {\mathfrak {G}}}$ ${\displaystyle g}$ ${\displaystyle A}$ ${\displaystyle (A;\parallel g)}$

Incidence set-theoretic

• The set of points that intersect with a certain straight line is called the support set of the straight line, this set is often noted as. Formalized: ${\displaystyle g\in {\mathfrak {G}}}$ ${\displaystyle g^{\circ }}$ ${\displaystyle g^{\circ }=\{A\in {\mathfrak {P}}:AIg\}}$
• From the first two axioms it follows that two straight lines coincide if and only if they intersect with the same points, that is, if their sets of carriers are equal. For this reason, in recent literature often equal assumed that every straight the amount of inzidierenden with her points is so . Then if and only if applies and the incidence relation can be completely replaced by the set-theoretic contain relation. ${\displaystyle g=g^{\circ }}$ ${\displaystyle AIg}$ ${\displaystyle A\in g}$

Levels

• From the third axiom it follows that every straight line is parallel to itself, from the fourth it follows that straight lines that are parallel and have a point in common are identical. In other words: if two lines are different and parallel, then they are disjoint.
• In general, disjoint lines do not have to be parallel.
• With the help of the fourth axiom, the fifth axiom can be formulated as follows:

If a triangle is given, and two points , and in such a way that the straight line parallel to the straight line is then intersect and . ${\displaystyle ABC}$ ${\displaystyle A'}$ ${\displaystyle B'}$ ${\displaystyle AB}$ ${\displaystyle A'B'}$ ${\displaystyle (A';\parallel AC)}$ ${\displaystyle (B';\parallel BC)}$

• In particular, the point is uniquely determined from the fifth axiom. ${\displaystyle C'}$
• If there is now a triangle, i.e. three points that do not lie on the same straight line, then the fifth axiom can be used to define a meaningful concept of a plane that is determined by the triangle . One possible definition is as follows: A point lies in if and only if the straight line intersects and . ${\displaystyle \varepsilon (ABC)}$ ${\displaystyle ABC}$ ${\displaystyle D}$ ${\displaystyle \varepsilon (ABC)}$ ${\displaystyle (D;\parallel BC)}$ ${\displaystyle AB}$ ${\displaystyle AC}$

• From the fifth axiom one can now show (with some technical effort and several case distinctions) that the following applies to lines that lie in: If two lines of the plane are disjoint, then they are parallel. Thus these “levels” fulfill all axioms of an affine level . ${\displaystyle \varepsilon (ABC)}$

In summary:

• An affine geometry that also includes the axiom of richness

"There are three different points (a" triangle ") that are not all on a straight line ." ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle {\mathfrak {G}}}$

satisfies, contains a plane , so that the points on this plane (as a set of points ) with their connecting lines ( as a set of lines ) with the restricted parallelism ( ) satisfy the axioms of an affine plane . ${\displaystyle \varepsilon }$ ${\displaystyle {\mathfrak {P}}_{\varepsilon }={\mathfrak {P}}\cap \varepsilon }$ ${\displaystyle {\mathfrak {G}}_{\varepsilon }=\{AB:(A,B)\in {\mathfrak {P}}_{\varepsilon }\times {\mathfrak {P}}_{\varepsilon }\;\land A\neq B\}}$ ${\displaystyle g\parallel _{\varepsilon }h\Leftrightarrow g\parallel h\land g,h\in {\mathfrak {G}}_{\varepsilon }}$

• Iff also applies when there is no so four are points that do not lie on a common plane is affine geometry affine plane. ${\displaystyle {\mathfrak {P}}_{\varepsilon }={\mathfrak {P}}}$

Examples

• Affine spaces generated by vector spaces over a body :
  • The Euclidean visual space can be generated by a three-dimensional vector space via . ${\displaystyle \mathbb {R} }$
  • The Euclidean plane can be generated by a two-dimensional vector space via . ${\displaystyle \mathbb {R} }$
  • Trivial examples are:
    • No point, no straight line ( -dimensional affine geometry) ${\displaystyle -1}$
    • a single point and no straight line (zero-dimensional affine geometry),
    • a straight line on which all points lie (one-dimensional affine geometry),

A geometry that is at most zero-dimensional can be viewed as a vector space over any arbitrary body, so it is also an affine space of the same dimension. From each set M which contains at least two elements, one can make one-dimensional affine geometry . This can be viewed as an affine straight line over a body if the body can be mapped bijectively onto . ${\displaystyle {\mathfrak {P}}=M;{\mathfrak {G}}=\{M\},\parallel =\{(M,M)\}}$ ${\displaystyle M}$

• The smallest affine geometry that contains a plane is the affine plane, which can be generated by the two-dimensional vector space over the finite body . It consists of the points and the straight lines , the connecting straight lines consist of exactly the two specified points. Furthermore applies . → See also the figures in affine plane . ${\displaystyle \mathbf {F} _{2}}$ ${\displaystyle {\mathfrak {P}}=\{A,B,C,D\}}$ ${\displaystyle {\mathfrak {G}}=\{AB,AC,AD,BC,BD,CD\}}$ ${\displaystyle AB\parallel CD,AC\parallel BD,AD\parallel BC}$

Desargue and non-Desargue geometries

All affine geometries generated by vector spaces over a body and even all affine geometries generated in the same way by left vector spaces over a skew body satisfy the great affine theorem of Desargues , they are affine spaces in the sense of linear algebra. The converse also applies to at least three-dimensional affine geometries: They can always be described by left vector spaces over an inclined body. But there are also planar affine ("non-Desarguessian") geometries (→ see affine plane ) that do not satisfy Desarguessian theorem . They cannot therefore be generated by a vector space. Instead, you can always assign a ternary body to them as a coordinate area .

Embedding problem and coordinate areas

An affine space (in the sense of linear algebra) is always defined together with its coordinate range, a (skew) body and a - (left) vector space (with the exception of the empty affine space, which is still a subspace of a certain space is viewed as a sloping body). In linear algebra one usually restricts oneself to vector spaces over commutative fields, but the essential geometrical facts (apart from Pappus' theorem ) also apply more generally to left vector spaces over skew bodies. ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle V}$

This means that for affine spaces:

1. The space has a certain dimension , that is the dimension of the vector space. Additional definition: The empty set has the dimension . ${\displaystyle n=\mathrm {dim} (A)}$ ${\displaystyle -1}$
2. Due to the algebraic structure of the vector space, it is clear for each dimension what the structure- preserving self- mappings are: They can be described as affinities mainly through the structure-preserving mappings of the vector spaces. If one takes into account only the incidence structure and not the vector space structure, then one arrives at the larger group of (true-to-plane) collineations , which, however, can also be represented by affinities and body automorphisms for at least two-dimensional affine spaces.
3. For every smaller dimension than there are affine subspaces to which a - dimensional - subspace can be assigned. ${\displaystyle m}$ ${\displaystyle n\;(-1<m<n)}$ ${\displaystyle B\subset A}$ ${\displaystyle m}$ ${\displaystyle K}$
4. Each larger dimension can be understood as a subspace of an affine space of the dimension to which a -dimensional vector space is assigned (embedding). ${\displaystyle M>n}$ ${\displaystyle A}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle K}$

For affine geometries the following applies:

• If the geometry contains a plane , but does not coincide with it, then it is desargue and due to its incidence structure and its parallelism a definite oblique body and a definite dimension (at least 3) is given for the "coordinate vector space". ${\displaystyle K}$ ${\displaystyle K}$

• If the geometry is a plane that satisfies Desargues' theorem, the same is true with dimension 2.

It is precisely in these cases that the terms affine geometry and affine space become synonymous. In points 1 to 4 one simply takes over the terms of linear algebra.

• A non-Desargue plane also determines a unique coordinate structure, a ternary body , which, however, generally has much weaker properties than a sloping body.

1. The dimension of the geometry is by convention 2, because the geometry has more than one straight line (therefore more than one-dimensional) and disjoint straight lines are always parallel (therefore less than three-dimensional).
2. The structure-preserving images are true-to-the-line (and thus, in the flat case, trivially also true-to-parallel) bijective self-images of the plane, the affine collineations .
3. Every straight line of the plane is a subspace and of course a one-dimensional affine geometry, the one-point subsets are 0-dimensional subspaces.
4. Embedding in a geometry with a higher dimension is impossible.

• For zero and one-dimensional geometries that do not appear as subspaces of at least two-dimensional geometries, a structural investigation is obviously of no interest: Due to their incidence structure, nothing is given beyond the pure point set.

A generalization of the term affine geometry is the term weakly affine space . Every affine geometry is also a weakly affine space. Some nondesargue affine planes are real subspaces of weakly affine spaces, although such planes can never be embedded in more comprehensive affine geometries .

literature

• Günter Ewald: Geometry . An introduction for students and teachers (=  modern mathematics in elementary representation; 14 ). Vandenhoeck & Ruprecht, Göttingen 1974, ISBN 3-525-40536-7 (Table of contents: DNB 750005521/04 [accessed December 25, 2011] American English: Geometry, an introduction . Translated by Anneliese Oberschelp).
• Rudolf Fritzsch: Synthetic embedding of Desarguessian levels in rooms . Mathematical-Physical Semester Reports, No. 21. 1974, p. 237–249 ( (still without full text) ).
• Jeremy Gray : Worlds out of nothing: a course of the history of geometry of the 19th century . 1st edition. Springer, Berlin / Heidelberg / New York 2007, ISBN 978-0-85729-059-5 .
• Günter Pickert : Projective levels (=  the basic doctrines of the mathematical sciences in individual representations with special consideration of the areas of application. Volume 80 ). 2nd Edition. Springer, Berlin / Heidelberg / New York 1975, ISBN 3-540-07280-2 .
• Daniel Richard Hughes, Fred. C. Piper: Projective planes . Springer, Berlin / Heidelberg / New York 1973, ISBN 0-387-90044-6 .

References and comments

1. ↑ ^{a b} Ewald (1974)
2. ^ Gray (2007)
3. ↑ One can show that for a space with more than 2 points on every straight line, every bijective self-mapping, i.e. every collineation, is at the same time true to the plane. For rooms with exactly two points on each straight line, the level accuracy must also be required. This special case is presented in detail in the article " Collineation ".
4. ↑ ^{a b} Pickert (1975)
5. ↑ Fritzsch (1974)