5.3. Affine and vector spaces
This page is under construction.
Let us now apply the tools just introduced to two geometries (describing
distinct but related categories of
spaces), first affine geometry, whose
models are called affine spaces and morphisms are the affine
maps, then to the theory of vector spaces we shall call linear geometry,
whose morphisms are the linear maps. The automorphism group of an affine
space is called its affine group. The study will first focus on finitedimensional spaces
(infinite dimensional ones are harder to classify, and require to specify the
geometry among nonequivalent possible formalizations).
Finite dimensional spaces of each kind (affine and linear) will fall into one
isomorphism class for each dimension, which can be any n∈ℕ,
and all their subspaces (images of morphisms) are sections (thus spaces in their own right).
Affine subspaces and directions
In affine geometry, invariant structures in spaces E with dimension n = 2
or more, can be defined from the notion of their subspaces, which are intuitively
understood as straight (and limitless, for those with nonzero dimension).
Affine subspaces are classified into types E_{p}
by their dimension p between 0 and n ; each E_{p} is the set of images of injective affine maps from
pdimensional spaces to E :

E_{0} = E as zerodimensional subspaces are the singletons ;
 E_{1} is the notion of straight line, which can be simply translated into
the 3ary relation of alignment between points ;
 E_{2} is the notion of plane in E ;
 E_{n−1} is the notion of hyperplane ;
 E_{n} = {E}, not structuring.
Equivalently, affine subspaces are the preimages of points by affine maps; pdimensional
subspaces are the preimages of points by surjective affine maps to (n−p)dimensional
spaces.
From this we can define an equivalence relation on each E_{p} : two
pdimensional subspaces are parallel if they are either equal or disjoint
but both included in a common (p+1)dimensional subspace.
Equivalently, they are parallel if there is an affine map from E, by which they are
the preimages of 2 points, i.e. they belong to a common partition of E by an affine map.
Let us call affine quotient such a partition.
In affine geometry, parallelism is an equivalence relation: for each subspace
there exists a unique affine quotient to which it belongs, that is called its direction.
So "to be parallel" can be phrased "to be in the same direction".
Intuition of affine geometry
Another invariant structure intuitively summing up affine geometry in dimension 2 or more,
is the notion of parallelogram, conceived as 4tuples (to directly fit in this
presentation; their view as whole figures with sides and inside, definable using the
order in ℝ, is also an affine invariant), and similarly parallelepipeds of any dimension.
Here are intuitive descriptions of affine maps:

An affine map between planes can be intuitively understood
as the approximation of a transformation obtained by looking at a figure on a plane in
perspective from a large distance, and eventually through a mirror. Indeed this is the composite
of two affine maps: one from a plane to the space, followed by one from that space to
another plane (taking a picture of a region of the space from a viewpoint whose
distance to the object is much larger than its size, as much as needed for the desired accuracy).

In a 3dimensional space, a projection to a plane in parallel to a line intersecting this plane,
is an affine transformation, which is a limit of ( = can be
approached by) automorphisms that shrink and progressively "smash" the space
onto the plane.
As any function is surjective to its image (which is here a subspace), any affine map
is a composite of a surjective one (a quotient) with an injective one (isomorphism to a
subspace).
More structures of affine geometry (also preserved by its automorphisms),
will be introduced later:
 Vectors, i.e. translations, forming a subgroup of the monoid of
affine transformations
 Ratios of volumes (thus "areas" in planes), defined by counting numbers of
small parallelepipeds which can fill them; ratios of kdimensional volumes in parallel kdimensional
subspaces (0<k<n).
 In planes: notions of ellipse, parabola, hyperbola; center of an ellipse or
hyperbola (but no data on foci, axis and eccentricity). Similar notions exist in higher dimensions.
More detailed study of affine geometry
in another page.
Up: 5. Geometry
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