3
Group of translations:
Consider 2D space (x,y). In such space, a translation is defined by a translation vector (Dx, Dy). We usually write:
(27)
x' = x + Dx
y' = y + Dy
To obtain the new values x' and y', we use addition. Could we get the same results through multiplication?
Consider the following matrices:
(28)
Note that they are defined by two independent parameters Dx and Dy. Therefore, the dimension of the group is 2.
Form:
(29)
Note that this is fundamentally different from simple matrix multiplication
(30)
g × r
It is a specific group action.
(31)
Additionally, we can consider translations in 3D or 4D spaces. The corresponding square matrices, forming groups, are:
(32)
(33)
The corresponding action is:
(34)
The group of translations is commutative. Its neutral element is the null translation.
Groups of matrices: why?
With matrix groups, we can combine several operations into one, into a single action. Consider the following matrices and the following action:
(35-1)
We combine two things: a rotation (angle a), plus a translation (Dx, Dy).
The element g of the group G acts on the space r = (x,y), not "directly", but through a more refined "action".
This group
(35-2)
called "Special Euclidean group SE(2)", acts on 2D space. This name will be explained further.
What is its dimension? It depends on three free parameters: (a, Dx, Dy), so its dimension is three. We may write:
g ∈ SE(a, Dx, Dy)
Subgroups.
For us, a group is a set of square matrices. Among this set we can find subsets.
g ∈ SE(0, Dx, Dy) is the subgroup of translations.
g ∈ SE(a, 0, 0) is the subgroup of rotations around the origin 0.
g ∈ SE(0, Dx, 0) is the subgroup of translations parallel to the axis OX.
The above group carries points. These points have no special characteristics. They are just... points, nothing else.
But later, other groups, which describe the physical world, will carry points which have different characteristics, "attributes": mass, energy, momentum, spin...
With the above group, only sets of points are interesting to carry. Here appears the fundamental concept of:
Species.
Our first group carries geometric objects, which are sets of points, geometric ("rigid") figures. The simplest set is composed of two points. Consider pairs of points in a 2D space:
(35-3)
In figure (35-3), two pairs of points (A,B) and (A',B') have been drawn. I can find an element of the group that transforms (A,B) into (A',B'): by combining a rotation around the point O and a translation. See figure (35-4).
(35-4)
Now consider the two pairs:
(35-5)
It is impossible to find any element g (square matrix) of my group G which can carry (A,B) onto (A",B"). I will say that:
(A,B) and (A',B') belong to the same species.
(A,B) and (A",B") belong to different species.
The characteristic of a species of pairs of points is called length.
This is the definition of length in terms of group theory.
How can you assert that two segments have the same length? Because you can compare them, by superimposing one onto the other.
In our group, two segments with different lengths belong to different species, because our group does not allow dilations or contractions (homothetic transformations). The group that deals with this is a different one ("Special Cartesian group"):
(35-6)
With respect to this group, all pairs of points belong to the same species. The dimension of this group is four.
Instead of two points, we could consider three or four points, these last forming, for example, squares.
(36)
With respect to the group (35-1), squares whose sides have the same length belong to the same species. But if the sides of two squares are fundamentally different:
(37)
they belong to different species.
This group, which governs 2D translations and rotations around a fixed point of a plane, is the Special Euclidean group: SE(2).
We can easily imagine a similar group acting on a 3D space. The groups of 3D and 4D translations were given in (32) and (33).
We can easily imagine a group describing translations in an n-dimensional space. But what about rotations?
We can imagine a rotation in a 3D space. We can even write it using a matrix containing three angles, the Euler angles: its dimension is therefore three.
Index Dynamic Groups Theory