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,c*
∈*E*, that we shall denote (*a*−*b*−*c*).

**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:

- (
*a*−*b*−*c*) ⇔ (*b*∈[*ac*] or*a*∈[*bc*] or*c*∈[*ab*]). - (the definition of betweenness from alignment would be more
complicated, and will not be discussed here)

- Alignment is definable from straight lines, by

∀*a,b,c*∈*E*, (*a*−*b*−*c*) ⇔ (∃*l*∈*L*,*a*∈*l*∧*b*∈*l*∧*c*∈*l*). - Conversely, the type
*L*is constructible using alignment, as follows.

∀*a,b*∈*E*,* a*≠*b* ⇒ ∃!*l*∈*L*, *a*∈*l*
∧*b*∈*l*

so that we have a partial operation that for any 2 points *a*≠*b*,
gives the line *a*∨*b* containing them : it is the
function from {(*a,b*) ∈* E**ï¿½**E*| *a*≠*b*}
to *L*, defined by

∀*a,b*∈*E*,* a*≠*b* ⇒ (∀*l*∈*L*, *l*=
*a*∨*b* ⇔ (*a*∈*l* ∧ *b*∈*l*))

Moreover, this operation needs to be surjective : ∀*l*∈*L*,
∃*a,b*∈*E*,* a*≠*b* ∧ (*l* = *a*∨*b*).

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* = {*a*∨*b*|*a,b*∈*E*,* a*≠*b*
} where *a*∨*b*={*c*∈*E*|(*a*−*b*−*c*)}

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.

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*).

**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 *F* ⊂ *E* is called an hyperplane if it
satisfies the following equivalent conditions

*S*^{+}(*F*) = {*F*,*E*} and*F*≠*E*- ∃
*a*∈*E*\*F*,*E*=*F*∨*a* - Dim
*F*=*n*−1

((S^{+}(F)
= {F,E}) ∧ (F≠E)) |
⇔ ((∃a∈E, a∉F) ∧ (∀a∈E\F,
E=F∨a)) |

⇒ (∃a∈E\F, E=F∨a)
⇒ (Dim F ≥ n−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),

∀Now let us see how the data of the setF∈S(E), ∀a∈E\F, Dim(F∨a) = Dim(F)+1.

For every subset

When

Namely, it is similar to the formula

We can restore 〈

∀

proven as follows:

We can build a sequence of subspaces

F=F⊂...⊂_{k}F_{n-1}=h

with (for everyifromkton-1) DimF=_{i}iandx∉Fas follows:_{i }

Given such anFwith_{i}i<n-1, choose a pointp∉_{i}F∨_{i}x.

(This exists because Dim(F∨_{i}x)<n)

Then we defineF_{i+1}=F_{i}∨p._{i}

Thus, DimF_{i+1}=i+1

Also,Dim (F_{i}∨p∨_{i}x) = 1+Dim(F∨_{i}x) =i+2

thusx∉F_{i+1}.

(For infinite-dimensional spaces, there is also a proof using the axiom of choice).

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 *= Dim*G*)∧(*F*
ǀ⎜*G* ))

The space Par(*F*,*a*) can also be defined as {*b*∈*E*|*
b*∨*a* ǀ⎜*F*}

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 |
||

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).

- 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).

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

**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.

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. |
Aï¿½ï¿½ï¿½ï¿½D / / / I / / / f(A)ï¿½ï¿½ï¿½f(D) |

The action of a translation on a point is denoted as an "addition" between the point and the vector:

For any two points

We might also write = B−A

The application of a vector to another point

As this vector is also the only translation that sends

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

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*(*b***v**) = (*ab*)**v** - ∀
*a*,*b*∈ℝ, ∀v∈V, (*a*+*b*)**v**=*a***v**+*b***v**

- ∀
*a*∈ℝ, ∀u,v∈V,*a*(**u**+**v**) =*a***u**+*a***v**

- ∀u∈V, 1
**u**=**u**and 0**u**= 0

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

- 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*+*A.*

- 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 ℝ)^{n} - 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