Affine geometry

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.

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) ∈ EE| 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.

Affine Subspaces

Definition. A subset F of an affine space E is said to be a subspace of E, 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).

Any subspace F of an affine space E is also an affine space, where the alignment relation, as well as the operation ab giving the line through points ab in F, are defined by the restriction of those in E. The set LF of straight lines in F is defined from the set L of those in E, as LF = {lL|lF}.

Subspaces are classified by their dimension kn. Namely, they are
A formal definition of the dimension is given below.
By the singleton operator that injects E into S(E), we can see E as a subset of S(E).

Intersection of subspaces. The intersection of any (nonempty) set of subspaces of an affine space, is also a subspace.

(The proof is immediate from the definition.)

For any family of subspaces (Fi)iI of E where I≠∅, we have S(⋂iIFi)=⋂iI S(Fi) so that the operation  ⋂iIFi on the family of subspaces Fi, first defined as the intersection of the Fi seen as sets of points, may as well be interpreted as an intersection on their sets of subspaces.

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).

Dimension of a subspace. A nonempty subspace of an affine space is said to have dimension k∈ℕ, if it is generated by a set of k+1 points but is not generated by any set of k points or less.

(In particular this defines the dimension of the space E itself)

The operation ab of giving the line through given points, is generalized as

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).

Hyperplanes.

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).

Parallel subspaces

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 FS(E)\{∅} and any point aE, the parallel subspace to F through a, is defined as

It has the following properties:

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), (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}

Affine transformations

Like with any systems, we have a notion of isomorphism between affine spaces.
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 affine transformations, such that any composite of affine transformations is also an affine transformation.

Let us intuitively describe the non-bijective affine transformations:
The typical and most common example of surjective affine transformation, is the one with n=3 and p=2, that consists in taking a picture of a region of the space from a faraway viewpoint (whose distance is approximated as infinite compared to the size of the pictured object).

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 pn, any direction of (n-p)-dimensional subspaces has a natural structure of p-dimensional affine space, such that ....

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 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:

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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