(For a short simplified version without reference to the
prerequisites, just read the text in green)
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
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)
|(x , y+a)
(x+a , y)
(its coordinates = (0,0))
|Shear mapping w.r.t. the horizontal axis
||(x+a.y , y)
|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
("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, but the following ones.
Some fundamental affine structures (each of which suffices to define all others), are
This list of structures still applies in higher dimension
except that ratios of areas only apply to parallel planes;
instead, we have ratios of n-dimensional volumes.
(Precisely, for any 0<k<n we can also compare the k-dimensional volumes in parallel k-dimensional subspaces, so that the ratio of lengths is the particular case k=1, and the comparison of volumes is the case k=n where they are always parallel).
This study can be generalized to
higher dimensions : for each value of the dimension n
there is one n-dimensional affine geometry, with only
one isomorphism class of models (n-dimensional affine
spaces), and an
affine group for each model.
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).
More detailed study of affine geometry
in another page.
Some geometric spaces, such as vector spaces, Euclidean spaces,
and both space-times without gravitation: the classical one (the
Galilean space-time), and the one of Special Relativity (the
Minkowski space), are "richer" than affine geometry, i.e.
"contain an affine structure", in the following (roughly)
But other geometries do not contain an affine structure, either
because they have no notion of straight line, or because these
lines do not satisfy the axioms of affine geometry. For example
Like affine spaces, projective spaces are also based on the structure of alignment, but have no other affine structures from the above list. Parallels cannot be defined there as any pair of straight lines in a plane has one intersection point. Such spaces can be described as affine spaces together with extra points "at infinity" in the role of intersection points of parallel lines, while ignoring which points are at infinity and which are not. These points "at infinity" form the line of "horizon" that is a straight line like others. Projective transformations of the plane (automorphisms of the projective plane) are those involved for perspective representation. Among them, affine transformations are those keeping the horizon to itself, so that affine geometry is equivalent to projective geometry with a constant symbol named "horizon" with type "straight line".
It has a notion of circle (or sphere) but cannot distinguish
straight lines among them.
Follow the link for details.
Still, affine geometry is not far away from the above, as small regions of these spaces are approximately affine too: the smaller a region, the better its description can be approximated by affine geometry (with possibly more structures). Usual geometric spaces will have this property of being approximately affine in small regions : we will say they are smooth.
Differential geometry is the "geometry" whose only structure is
the notion of smoothness, and smooth curves.
In particular, smooth spaces have an approximation for ratios of small volumes as they become smaller and closer to each other.
The smoothness structure cannot be restricted to a structure of
"being approximately projective in small regions" because the
"approximately affine structure" of small regions would anyway be
definable from it, as the horizon relatively to a small region,
has its needed approximative definition as "what is not near this
This is even weaker than differential geometry, as it has a
notion of curve requesting their continuity, that is less
restrictive than smoothness. For example the Koch
snowflake is continuous but not smooth, so that it is
distinguished from curves by differential geometry, but not by
Introduction to topology
Let us first only present the structures, assumed to obey the
properties of the intended spaces. The axiomatic
specification of these properties will only be completed
Euclidean geometry (with any
dimension >1) consists in affine geometry together with one
structure, thus named the Euclidean structure, that can be
expressed (represented) in the following equivalent forms (any
one suffices to define the others) :
geometry may also admit one more structure from the previous
list : the unit of area (or unit of volume, in the case
of higher dimension), which finally (thanks to the above Euclidean
structure) can be called a choice
of unit of distance. The choice to include or not this
structure in the definition of "Euclidean geometry" is debatable,
with motivations for or against it, as follows
For any dimension, the operation
of distance d(A,B) between any two
points A and B (with values either among
real numbers or in a set of quantities, depending on the above
choice) can be seen as the
fundamental structure of Euclidean geometry, as it is completely
sufficient to define all other structures of this geometry in a
rather natural way.
Indeed, distance suffices to define the affine structures: the
betweenness relation is defined as
B∈[AC] ⇔ d(A,C) = d(A,B) + d(B,C)
and others can be
defined from it. Therefore, the transformations preserving
distance (even if its values are mere quantities), called isometries,
preserve all other structures of Euclidean geometry as well.
Other presentations of the Euclidean structure, assume a priori
As for the notions of circle (or sphere) and intersection angles,
they may suffice to define affine geometry but only
The details of these correspondences between different formulations of the Euclidean structure, will be explained in the introduction to inversive geometry.
The isometries of an Euclidean plane or space are called
moves (to be distinguished from the isometries of other
spaces such as a sphere, with also an operation of distance but
that does not satisfy all the same axioms). The space of
isometries of the Euclidean plane is 3-dimensional, and split in 2
pieces (each of which is also 3-dimensional):
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