Affine geometry (continued)

Straight lines and the alignment relation

An affine space with dimension n ≥ 2 is a set E of points (i.e. affine geometry is a theory with one basic type E), with one structure that may equivalently be expressed in 3 ways:

Betweenness: the 3-ary relation b∈[ac]

Alignment is another 3-ary relation between points a,b,cE, that we shall denote (abc).

Straight line is a second-order structure, that is a type L of objects related to points by a relation usually denoted ∈ (as lines are usually conceived as sets of points) called the incidence relation between points and (straight) lines.

(It would be also possible to conceive points as sets of lines, i.e. conceive points as a second-order structure over lines, as the incidence relation also separates points).

Each one suffices to define the others:

For this to be possible, it needs to satisfy the axiom

a,bE, ab ⇒ ∃!lL, albl

so that we have a partial operation that for any 2 points ab, gives the line ab containing them : it is the function from {(a,b) ∈ E�E| ab} to L, defined by
a,bE, ab ⇒ (∀lL, l= ab ⇔ (albl))

Moreover, this operation needs to be surjective : ∀lL, ∃a,bE, ab ∧ (l = ab).

In these conditions, the above definition of alignment from straight lines, can be reversed as the construction of L from alignment expressed in the set theoretical formalism by
L = {ab|a,bE, ab } where ab={cE|(abc)}

If we want a formalism purely based on the alignment predicate, then it needs the axioms obtained by re-expressing the above axioms on L and the definition of alignment from L, as formulas purely about alignment, by interpreting L as constructed from alignment (thus an abbreviation of its use) in the way we last expressed.

A subset F of an affine space E is a subspace of E, if and only if ∀a,bF, ababF.


The set of all subspaces of E will be denoted by S(E). For every subset FE, the set {GS(E) | FG} of subspaces of E containing F , will be denoted as S+(F).

Subspace generated by a set. For any subset F of an affine space E, the subspace generated by F is defined by 〈F〉 = ⋂S+(F) (the intersection of all subspaces of E containing F). It is the smallest subspace of E containing F for the inclusion order (it is one of them and included in any other).

In any n-dimensional affine space E (with n>0), a subspace FE is called an hyperplane if it satisfies the following equivalent conditions

However, their equivalence, satisfied in affine geometry, cannot (?) be deduced from the above formulas (axioms and definitions). Instead of equivalences, they only imply that
((S+(F) = {F,E}) ∧ (FE)) ⇔ ((∃aE, a∉F) ∧ (∀aE\F, E=Fa))

⇒ (∃aE\F, E=Fa) ⇒ (Dim Fn−1)

Now let us accept these equivalences (as we did not yet fully formalize affine geometry), including for the affine geometry of any subspace with dimension >1. In particular (with separate justifications for dimension <2),
 ∀FS(E), ∀aE\F, Dim(Fa) = Dim(F)+1.
Now let us see how the data of the set E* of all hyperplanes of E, suffices to defined the set S(E) of all subspaces of E (instead of using the set L of lines), and thus all the affine geometry itself.

For every subset FE, let F= E*∩ S+(F) = {hE*|Fh}={hE*|∀xF, xh}, which only depends on 〈F〉 (because ∀hE*, Fh ⇔ 〈F〉⊂h).

When F is a subspace, this definition is just a reinterpretation of F as a set of hyperplanes, that is symmetric to the basic conception of F as a set of points, when reversing the inclusion order in S(E).
Namely, it is similar to the formula F=ES(F) = {xE|{x}⊂F}.

We can restore 〈F〉 from F by 〈F〉 = ⋂F= {xE|∀hF, xh}. For this to be true, affine geometry needs to satisfy the formula

FS(E), ∀xE\F, ∃hE*,Fhxh.

proven as follows:
We can build a sequence of subspaces
F= Fk⊂...⊂Fn-1=h
with (for every i from k to n-1) Dim Fi=i and xFi
as follows:
Given such an Fi with i < n-1, choose a point piFix.
(This exists because Dim(Fix)<n)
Then we define Fi+1 = Fipi.
Thus, Dim Fi+1=i+1
Also,Dim (Fipix) = 1+Dim(Fix) = i+2
thus xFi+1.
(For infinite-dimensional spaces, there is also a proof using the axiom of choice).
Subspace generated by a family of subspaces. For any family of subspaces (Fi)iI of E, the subspace generated by the Fi is defined by ⋁iI Fi = 〈 ⋃iI Fi

Remark. S+(⋁iI Fi)= ⋂iI S+(Fi).

Affine lines

The data of the sets of subspaces were sufficient structures for affine spaces with dimension >1 but not yet for 1-dimensional affine spaces (lines in themselves outside the context of a larger dimensional space), whose set of subspaces is trivial (not structuring) and thus not a sufficient structure to describe 1-dimensional affine geometry with its "small" set of automorphisms (that are not all permutations).

Other concept of parallel spaces

For any FS(E)\{∅} and any point aE, the parallel subspace to F through a, is defined as ∀GS(E),∀aE, (Par(F,a)=G) ⇔(aGFG), or

We can also define a more general concept of parallelism that is only a preorder relation between nonempty subspaces F and G, named "F is parallel to G" and denoted F ǀ⎜G, that is related with the previous one by
(F ǀ⎜G )⇔(∃F'S(E), (FF')∧(F'G))⇔ (∃G'S(G), FG')⇔(∀aF, F⊂Par(G,a))
⇔((FG)∨((FG = ∅)∧(Dim(FG) = Dim(G)+1)))

In other words, while the condition FG = ∅ does not generally suffice to make FG, it still suffices to make F ǀ⎜G when G is an hyperplane of the space FG.

Also, FG ⇔ ((Dim F = DimG)∧(F ǀ⎜G ))

The space Par(F,a) can also be defined as {bE| ba ǀ⎜F}

Structures and permutations in the plane

In the following table, every permutation in a line (named then defined using coordinates, with an amplitude depending on a parameter a∈ℝ), preserves (is an automorphism for) every structure named in a different line (in the last column).
The third column counts the dimension of the permutation group described in each line.
For the sake of naming things, the plane is thought of as a human-size vertical plane with a north-south direction, crossed by the Earth's equator. The coordinates are (x,y) where x = latitude, and y = altitude.

Permutations names Image of (x,y)
Vertical translation
Horizontal translation
(x , y+a)
(x+a , y)
Origin (constant point).
(its coordinates = (0,0))
Shear mapping w.r.t. the horizontal axis
(x+a.y , y)
Euclidean structure:
circularity, angles...
Squeeze mapping w.r.t. the vertical and horizontal axis (x/a, a.y) with a>0
Rotation around origin (x.cos a y.sin a, y.cos a + x.sin a) 1
Altitudes comparison
("to be higher than")
Dilation [from/to] origin (a.x, a.y) with a>0 1
Unit of area
Reflection w.r.t. vertical axis (in pair with Id) (-x , y) 0 Orientation (left/right, sign of angles)

Affine geometry, is the geometry whose structures (the affine structures), are preserved by all of the above listed permutations.

Thus, these do not include any of the above listed structures.

In dimensions higher than 2 we have the 4-ary relation saying that 4 points are in the same (2-dimensional) plane, and generally an (n+2)-ary relation between points to say that they belong to one n-dimensional subspace, but this can be defined out of the 3-ary relation of alignment.

Measurements of volumes in an n-dimensional affine geometry can be defined as an operation between n+1 points, with value a quantity: the volume of symplex or parallelepiped defined by these points (in plane geometry, a symplex is a triangle, and a parallelepiped is a parallelogram in 3-dimensional geometry, a symplex is a tetrahedron).

Properties the affine group

A transformation of a plane, is called an affine transformations if it satisfies the following properties, which are equivalent:

Let us further comment the above table of diverse affine transformations of the plane.

For every affine transformation f obtained as composite of any number of permutations from given lines of this table (with possible repetitions in any order), and for every (possibly different) choice of order between (only) these lines, there is a unique tuple of values of the parameters of permutations in these lines (except adding multiples of 2π for rotation angles) so that their composite in this new order (now without repetition) coincides with f. This tuple of values of parameters can be used as "coordinates" of f (their number is thus the dimension of the space of automorphisms for structures from the rest of lines).

For this, we had to put vertical and horizontal translations together, as they are mixed when composed with rotations, depending on the composition order. Similarly, shear mappings mix vertical translations with horizontal ones (though not vice versa), and rotations mix shear and squeeze mappings together.

The permutations in each line of the table, move the structure in the same line to "all its possible other values" (in the framework of affine geometry) without repetition. This way, a structure of a kind described in a given line of the table, can be chosen (added to the language, with a value among its "other possible values" from affine geometry) independently of choices of other kinds (described in other lines). It brings no information expressible by closed formulas; the only "effect" of a list of choices of structures from given lines of the table, is to reduce the automorphism group of the resulting geometry, to the mere set of composites of permutations from the complementary list of lines of the table.

There may be other ways to split the affine group as the set of composites from a list of subgroups (and each subgroup as a list of 1-dimensional ones) satisfying the above remarks (that it forms a sort of coordinates system for the group... ). The above way has 2 advantages:

An affine space with an origin, is a vector space.

Orientation : in an n-dimensional affine space, the set of (n+1)-tuples of points that belong to a common (n-1)-dimensional subspace, separates the set of all (n+1)-tuples of points in 2 parts. An orientation of the space consists in giving different names ("right" and "left") to these 2 parts.


The notion of vector can be formally defined as being another name for a translation.

We might for example define the concept of translation in this way : a transformation f is a translation if for any 2 points A and D, the segments [A f(D)] and [f(A)D] have the same middle I.
For any points A and B there exists one and only one translation that sends A to B: for any D, f(D) is the only point C that completes the parallelogram, such that [AC] and [BD] have the same middle.
In clear, the points A,B,C,D are vertices of a parallelogram if they are not aligned, and their configuration forms a smashed parallelogram if they are aligned.
    /                 /
   /       I       /
 /                 /

The action of a translation on a point is denoted as an "addition" between the point and the vector: B= v+A (because it is computed in any coordinates system, by an addition for each coordinate: each coordinate of B is the sum of that coordinate for A with the corresponding component of v).
For any two points A and B, we define the vector v=\scriptstyle \overrightarrow{AB} as the only translation that sends A to B (i.e. such that B= v+A). In the case of Euclidean plane geometry, it is usually drawn by an arrow from A to B.
We might also write \scriptstyle \overrightarrow{AB} = B−A

 The application of a vector \scriptstyle \overrightarrow{AB} to another point D, that is C= \scriptstyle \overrightarrow{AB} +D, is such that they form a parallelogram as described above.
As this vector is also the only translation that sends D to C, the arrow from D to C is another representation of the same vector.

The set of translations forms a group. The identity element is called the zero vector.
We define the addition of vectors to mean the composition of translations.
We have an operation of multiplication of a vector v by a real number a, which gives another vector: the one that moves in the same direction, a times further (thus, in the opposite way if a is negative).
Multiplying a vector by -1 gives its opposite (in the language of vectors), that is its inverse as a bijection (and as the inverse element in the group of vectors).
Denoting V the set of vectors, these operations satisfy the axioms of vector spaces:

Defining affine geometry

Affine geometry is the geometry that describes spaces with any number n of dimensions, with the following possible presentation :
We have 3 sets (types of objects)
related by the following list of structures (concepts), called the affine structures, which are contained in (but poorer than) those of the usual Euclidean geometry with that dimension, and with the same properties : A possible axiomatic expression of these properties consists in the following:
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