After our general introduction
to geometry,
let us more precisely introduce affine geometry, that is the
description of affine spaces (classified by their dimension).
There are diverse equivalent ways to formalize this as an
axiomatic theory. Some formalizations can even let the dimension
undetermined from 0 to infinity, thus showing what all affine
spaces have in common (why they deserve a common name), letting
separate axioms specify the dimension. While infinite-dimensional
affine spaces theoretically exist (as models of the general
theory), they have some subtleties beyond the scope of this
introduction, so here we shall only consider spaces with a finite
dimension *n*∈ℕ.

But the below presentation of affine geometry will be a mere
introduction by another formalism only working for *n*>1,
without giving any full axioms list (thus letting properties be
given by intuition). This approach is motivated both by
pedagogical reasons and its interesting way of introducing some
important mathematical concepts that also apply (slightly
modified) to the descriptions of other systems (projective spaces,
vector spaces...). It intuitively motivates the fully rigorous
studies of the work on algebra,
that is rigorously independent of the present section, and that
will ultimately provide its rigorous foundations.

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**×**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.

**Definition.** A subset *F* of an affine space *E*
is said to be a *subspace* of *E*, if ∀*a,b*∈*F*,*
a*≠*b* ⇒ *a*∨*b* ⊂ *F*. 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*).

Subspaces are classified by their dimension

- The empty set ∅, that (if we want a dimension to be given) has
dimension -1;

- Points, i.e. singletons {
*a*}⊂*E*, with dimension*k*=0;

- Straight lines
*l*∈*L*(with dimension*k*=1); - Planes (with dimension
*k*=2); - ...

- The whole space
*E*, with dimension*k*=*n*.

By the singleton operator that injects

(The proof is immediate from the definition.)

For any family of subspaces (

(In particular this defines the dimension of the space

The operation

*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).

In an affine plane, any two lines with empty intersection are
parallel. However this is not true in general. Already in
3-dimensional affine spaces there are lines with empty
intersection, which are not parallel (they have different
directions). Thus we need more subtle definitions for parallelism.

For any *F*∈*S*(*E*)\{∅} and any point *a*
∈*E*, the parallel subspace to *F* through *a*,
is defined as

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

It has the following properties:

- Par(
*F*,*a*)∈*S*(*E*)

- Dim (Par(
*F*,*a*))=Dim*F*

- ∀
*F*∈*S*(*E*)\{∅}, the binary relation on*E*defined by (*b*∈Par(*F*,*a*)), is an equivalence relation. The membership to the corresponding partition, defines the binary relation ∥ in*S*(*E*): ∀*G*∈*S*(*E*),∀*a*∈*E*, (Par(*F*,*a*)=*G*) ⇔(*a*∈*G*∧*F*∥*G*). This is the relation of parallelism*F*∥*G*is read "*F*and*G*are parallel". - This relation ∥ is itself an equivalence relation in
*S*(*E*), so that "to be parallel" can also be phrased "to have the same direction". Defining the direction of a subspace*F*as its equivalence class for the parallelism relation, it is itself a partition of*E*.

These properties of affine spaces are not consequences of the previous formulas (as hyperbolic geometry in any dimension presents a hierarchy of subspaces as we described but does not satisfy these properties of parallelism).

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*}

To specify its meaning, all we need is to specify the structures, or at least one or more structure(s) that suffice to define others. We gave 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).

Also, just as with homomorphisms of relational systems or algebras (and actually as a particular case of them for the structures that will describe affine spaces of any dimension), isomorphisms between affine spaces will just be the bijective cases in a larger set of functions between affine spaces called

Let us intuitively describe the non-bijective affine transformations:

- An injective affine transformation from an
*n*-dimensional affine space*E*to a*p*-dimensional space*F*is an isomorphism from*E*to an*n*-dimensional affine subspace of*F*. Thus it can only exist if*n*≤*p*(and it is an isomorphism if*n*=*p*). - Any surjective affine transformation
*f*from an*n*-dimensional space*E*to a*p*-dimensional space*F*, is such that the preimages of points of*F*form a direction of (*n*-*p*)-dimensional subspaces. (Thus it can only exist if*p*≤*n*, and it is an isomorphism if*n*=*p*). - Any other case is a composite of the above, as any function is
surjective to its image (which is here a subspace).

An example of affine transformation of the 3-dimensional space to itself, is given by the orthogonal projection to a plane in parallel to a line intersecting this plane. Note that this map can be approached by automorphisms that progressively "smash" the space onto the plane.

Conversely, for any

*This page is under
construction*

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).

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. |
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:

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