Affine geometry (continued)

Hyperplanes.

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

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 F ⊂ E, let F^⊥= E*∩ S⁺(F) = {h∈E*|F⊂h}={h∈E*|∀x∈F, x∈h}, which only depends on 〈F〉 (because ∀h∈E*, F⊂h ⇔ 〈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=E∩ S(F) = {x∈E|{x}⊂F}.

We can restore 〈F〉 from F^⊥ by 〈F〉 = ⋂F^⊥= {x∈E|∀h∈F^⊥, x∈h}. For this to be true, affine geometry needs to satisfy the formula

∀F∈S(E), ∀x∈E\F, ∃h∈E*,F⊂h ∧ x∉h.

proven as follows:

We can build a sequence of subspaces
F= F_k⊂...⊂F_n-1=h
with (for every i from k to n-1) Dim F_i=i and x∉F_ias follows:
Given such an F_i with i < n-1, choose a point p_i∉F_i∨x.
(This exists because Dim(F_i∨x)<n)
Then we define F_i+1 = F_i∨p_i.
Thus, Dim F_i+1=i+1
Also,Dim (F_i∨p_i ∨x) = 1+Dim(F_i∨x) = i+2
thus x∉F_i+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 (F_i)_i_∈_I of E, the subspace generated by the F_i is defined by ⋁_i_∈_I F_i = 〈 ⋃_i_∈_I F_i 〉

Remark. S⁺(⋁_i_∈_I F_i)= ⋂_i_∈_I S⁺(F_i).

Other concept of parallel spaces

For any F∈S(E)\{∅} and any point a ∈E, the parallel subspace to F through a, is defined as ∀G∈S(E),∀a∈E, (Par(F,a)=G) ⇔(a∈G ∧ F∥G), or

a∈F ⇒Par(F,a)= F
a∉F ⇒Par(F,a)= {a}∪((F∨a)\{b∨a|b∈F}).

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), (F⊂F')∧(F'∥G))⇔ (∃G'∈S(G), F∥G')⇔(∀a∈F, F⊂Par(G,a))
⇔((F ⊂ G)∨((F ∩ G = ∅)∧(Dim(F∨G) = Dim(G)+1)))

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

Also, F∥G ⇔ ((Dim F = DimG)∧(F ǀ⎜G ))

The space Par(F,a) can also be defined as {b∈E| b∨a 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)	Dim.	Structures
Vertical translation Horizontal translation	(x , y+a) (x+a , y)	2	Origin (constant point). (its coordinates = (0,0))
Shear mapping w.r.t. the horizontal axis	(x+a.y , y)	2	Euclidean structure: circularity, angles...
Squeeze mapping w.r.t. the vertical and horizontal axis	(x/a, a.y) with a>0	2	Euclidean structure: circularity, angles...
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:

It is an automorphism of affine geometry (i.e. it preserves the above affine structures)
It can be obtained as a composite of permutations from the above table
It can be written with parameters a,b,c,d,e,f such that a.e ≠ b.d, as
(x,y)↦(a.x+b.y+c , d.x+e.y+f)

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:

it shows the main diverse kinds of transformations
it includes the Euclidean structure.

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.

Vectors

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.

   A−−−−D
/                 /
   /       I    /
/                 /
f(A)−−−f(D)

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:

∀u,v,w∈V, u + (v + w) = (u + v) + w
∀u,v∈V, u + v = v + u
∀v∈V v + 0 = v
∀ a, b ∈ℝ, ∀v∈V, a(bv) = (ab)v
∀ a, b ∈ℝ, ∀v∈V, (a + b)v = av + bv
∀ a∈ℝ, ∀u,v∈V, a(u + v) = au + av
∀u∈V, 1u = u and 0u = 0

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)

We assume the set ℝ of real numbers to be known and fixed, the same for all affine spaces to consider.
A set P of points
A set V of vectors

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 :

We have an action + of the set of vectors on the set of points : ∀u∈V, ∀A∈P, u+A∈P
We have another operation of addition in V: ∀u,v∈V, u+v∈V
We have a multiplication of vectors by real numbers: ∀u∈V, ∀ a∈ℝ, au∈V (but other kinds of affine geometries may be defined taking other sets of numbers).
Whatever other structure that can be defined from the above. Example: for any 3 points A,B,C with A≠B, the alignment of the point C with A and B is defined by : ∃a∈ℝ, C = a $\scriptstyle \overrightarrow{AB}$ + A.

A possible axiomatic expression of these properties consists in the following:

The set V of vectors, with its operations of addition and multiplication by real numbers, is an ℝ-vector space with dimension n (i.e. isomorphic to the ℝ-vector space ℝⁿ)
The action + of the set of vectors on the set of points is a group action, and ∃ A∈P, ∀B∈P, ∃!v∈V, B=v+A (in other words, (v ↦ A+v) is bijective from V to P). Or equivalently, P≠∅ and ∀A,B∈P, ∃!v∈V, B=v+A

Note on the duality foundation

It lets all injective morphisms from finite dimensional spaces be sections, and all surjective morphisms onto finite dimensional spaces be retractions (which the algebraic foundation only ensures in finite dimensions).

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