What is geometry5.2. First-order invariants in concrete categories

The use of real numbers

Dimensions and semantic completeness

Multiple automorphisms and models5.3. Second-order invariants

The Galois connection (Aut, sInv)

The double meaning of "invariance"

Basis and algebraic structures

Endomorphisms5.4. Affine spaces

Parametered subspaces

Dual types

Subspaces, embeddings and sections

Quotients

Intuitive descriptions of affine maps5.5. Duality

Affine subspaces

Straight lines and algebraic structure

Directions

Duality structure

Other affine structures

Duality systems

Coordinate systems

...

The below is under construction

(For a short simplified version without reference to the
prerequisites, just read the text in green)

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)
equivalent senses:

- They contain, or at least can define and prove, the language and axioms of affine geometry
- They can be obtained as affine geometry with a choice of one
(or two...) more structure(s) that (as the dimension is finite)
is definable from affine geometry with a tuple of fixed
parameters among points (subject to a finite list of further
axioms)

- Their automorphism group is included in the affine group.

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

- Spheres (
*n*-dimensional spheres inside (*n*+1)-dimensional Euclidean spaces) and other curved spaces

- Automorphism groups of diverse spaces (affine, Euclidean and others)

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
small region".

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
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
later.

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) :

- The notion of circle, and (in
dimension >2) the notion of sphere

- The notion of rotation

- The measure of angles (between straight or intersecting lines, planes or spheres).
- Orthogonality (as a relation
between straight lines or intersecting lines)

- Ratios of distances, i.e.
distances with values in a type of quantities

- Data of the eccentricities, foci and axis of ellipses and hyperbolas
- Dot product between vectors (or inversely, between coordinates), with values in a type of quantities

Optionally, Euclidean
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 physics**, we have a different situation depending on the theory of physics being considered, and more precisely depending on the "framework vs specific theories" hierarchy. Namely,*as a framework*, Quantum Field Theory does not include a choice of unit of distance, but each specific substance (type of wave/particle) that it describes, usually comes with its own physical constant(s) (its mass and the intensity of interaction between substances) that break(s) the similarities between scales (making things behave differently at different distances). So, dilations may be automorphisms for the framework of quantum field theory itself, but not for specific laws of physics associated to specific substances inside this framework. The result is that there is not one best unit of distance for all situations, but there are diverse physical phenomena involving some physical constants, by which it is possible to absolutely express some possible choices for a unit of distance (such as the size of a given molecule, though this one may be too fuzzy to be used as a reference). In other words, there is not only one natural unit of length, but there are several ones available from diverse physical definitions.

As for General Relativity, it relates distances with densities, so that we may see it as dissimilar for distances insofar as we view densities as having an absolute unit. Of course, once put together, General Relativity and Quantum physics define an absolute unit of distance (the Planck length 1.616×10^{−35}metres) ; but this has no practical consequence.**For mathematics**, admitting both structures has the advantage that they are together expressible as one structure : an operation of distance, or equivalently a dot product, taking values in ℝ (instead of a type of quantities).

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
structures:

- Affine geometry is needed for introducing angles between straight lines (using the notion of straight line) or the dot product (using the notion of vector);
- The notion of smooth curve, thus some differential geometry, is assumed for introducing intersection angles.

As for the notions of circle (or sphere) and intersection angles,
they may suffice to define affine geometry but only

- either in a way that can be criticized as unnatural (as it
needs to distinguish straight lines from circles, which requires
measurements near infinity (to know "where the infinity point
exactly is"),

- or by using the ratio of volumes.

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
*Euclidean
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):

- Those preserving the orientation, are rotations and
translations. Indeed a rotation can be specified by its center
(dim=2) and its angle (dim=1). Or by choosing the image of a
given point (dim=2) and then the angle (dim=1). (Translations
are limit cases of rotations with small angles around faraway
points)

- Those reversing the orientation, are reflections with respect to any axis (= composites of rotations with the reflection by a fixed axis), composed with translations.

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