# Euclidean group

A movement is an isometric view of a Euclidean point space in itself. So there is a bijective, distance -preserving and conformal affine mapping.

Since the image of a geometric figure under such a figure is always congruent to the original figure, this is called a Motion also a congruence, this term is only in the case of a movement of two-dimensional Euclidean point space use.

From an actual movement is called, if the isometry additional benefit of getting the orientation. Otherwise, the motion is called improperly.

General certain bijections of the point space are characterized by axioms of motion as well as movements in the absolute geometry. You then define non-Euclidean geometries in the concept of congruence: Two figures are congruent if they are represented by a movement bijective on each other.

## Definition

Starting from a dimensional affine point Euclidean space and the scale on the Euclidean vector space called a moving image, if for any two points in

Here, the distance is defined as the length of its connection vector in

Such a map is automatically affine and bijective, ie affinity. She is also conformal.

## Description with the help of linear maps

Is a movement, then there exists an orthogonal map (linear isometry ) such that for all points apply:

So pick an origin, then for the position vectors of a point and its image point

Man that is the position vector of the image point obtained by the composition of the orthogonal Figure

And translation

## Description in coordinates

If we introduce in A an affine coordinate system with the origin and uses the associated base of the vector space, so you can choose any affine transformation by a matrix

And a translation vector

Describe:

Here are

The coordinate vectors of the position vectors and

When choosing a Cartesian coordinate system applies: if and only moving when the matrix is orthogonal, so true. Is in this case, it is an actual movement.

## The group of motions ( Euclidean group)

The sequential execution of two movements gives again a movement. Thus, the movement to form a group, the group or moving euclidian group which is designated or not. The sequential execution of two proper motions is again a real movement. Thus they form a subset of which is indicated by or. Both groups can be understood as the semi- direct product of the corresponding matrix groups or with the vector space. This means in practice that for the composition of two motions, and is:

Both groups are Lie groups of dimension

## Movements in the Euclidean plane

Are actual movements of the plane

- A parallel shift
- A rotation about a point in the plane a point reflection as a special case of a rotation of 180 °

Improper movements are

- One of the mirror
- A glide consisting of an axis of reflection followed by a translation along the axis.

The motion group ISO ( 2 ) of the plane can be generated by axis reflections.

## Motions in Euclidean space

Are actual movements in space

- A parallel shift
- A rotation about any axis in the space a reflection in a line as a special case of a rotation of 180 °

Improper movements are

- A plane reflection
- A glide, ie a plane mirror, followed by a translation in a direction parallel to the mirror plane
- A rotational mirror, so a reflection layer, followed by a rotation about an axis orthogonal to this plane,
- A point reflection

Rotations as well as rotary reflections always have fixed points. If one of the coordinate origin in such a, then the translational zero share. As stated in the article to orthogonal groups, has a rotation in space and always an axis and an angle of rotation is uniquely determined by these data. The same is true for three reflections.

In some situations, however, the translational part can not be omitted: for example, in the description of two rotations each other with non-intersecting axes.

The motion group ISO ( 3) of the space can be generated by plane reflections.

The motion of a rigid body in space, or even a camera movement can be seen as a continuous sequence of movements, ie, as an illustration of a real time interval in the group of actual movements of the space conceived.