Movement (math)

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In geometry , a movement is an isometric mapping of a Euclidean point space on itself. It is therefore a bijective , distance-maintaining and angular affine mapping .

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

One speaks of an actual movement if the isometry also receives the orientation . Otherwise the movement is called improper .

More generally, in absolute geometry , too, certain bijections of point space are characterized as movements by axioms of movement . They then define the concept of congruence in non-Euclidean geometries : two figures are congruent when they are bijectively mapped onto one another through a movement.


A mapping of the -dimensional Euclidean space in itself is called motion if it holds for two points and in

Here the Euclidean distance between the points and , i.e. the length of the line or the vector .

A movement is automatically affine and bijective, i.e. an affinity . In addition, it is conformal.

Description using linear illustrations

One can understand the -dimensional Euclidean space as an affine point space over a Euclidean vector space . Movements can then be described with the aid of linear algebra .

If there is a movement, there is an orthogonal mapping (linear isometry) , so that for all points and :

If one chooses an origin , then the following applies to the position vectors of a point and its image point

The position vector of the image point is thus obtained from the composition of the orthogonal mapping

and translation

Description in coordinates

If one introduces an affine coordinate system with the origin in -dimensional Euclidean space and uses the corresponding basis of the vector space  , then every affine mapping can be represented by an -Matrix

and a translation vector


Here are


the coordinate vectors of the position vectors and

When choosing a Cartesian coordinate system, the following applies: is a movement if and only if the matrix is orthogonal . If it is also true, then it is an actual movement.

A movement can also be formulated with the translation as the first action and the orthogonal mapping as the second action, because it is


The Movement Group (Euclidean Group)

The sequential execution of two movements results in a movement again. The movements thus form a group , the movement group or Euclidean group , which is denoted by or . Performing two actual movements one after the other is again an actual movement. These therefore form a subgroup of , which is denoted by or . Both groups can be understood as the semi-direct product or the associated matrix groups or with the group of translations. Specifically, this means that for the successive execution of two movements and the following applies:

Both groups are Lie groups of dimension

Movements in the Euclidean plane

Actual movements of the plane are

Improper movements are

The movement group ISO (2) of the plane can be generated by mirroring the axes .

Movements in Euclidean space

Actual movements in space are

  • a parallel shift
  • a rotation around any axis in space
    • a reflection on a straight line as a special case of a rotation by 180 °
  • a screwing , i.e. a rotation followed by a translation along the axis of rotation

Improper movements are

  • a plane mirroring
  • a glide mirror , i.e. a plane mirroring followed by a translation in a direction parallel to the mirror plane
  • a rotation mirroring , i.e. a plane mirroring followed by a rotation around an axis orthogonal to this plane
  • a point reflection

Rotations as well as rotational mirroring always have fixed points . If the coordinate origin is placed in such a system, the translational component becomes zero. As explained in the article on orthogonal groups , a rotation in space always has an axis and an angle of rotation and is clearly defined by this data. The same applies to rotating reflections .

In some situations, however, the translational part cannot be dispensed with: For example, when describing two rotations with axes that do not intersect one another.

The movement group ISO (3) of the room can be generated by plane mirroring .

The movement of a rigid body in space or a tracking shot can be understood as a continuous sequence of movements, i.e. as a mapping of a real time interval into the group of actual movements in space.


  1. The associative law is always fulfilled when executing one after the other.
  2. The figures and the splitting lemma
    are with than the unit matrix in and so that with applicable
    According to the multiplication rule   in   ,
    where it is carried out first , then what, however, has no influence on the subsequent term replacement , is therefore the one in the specification of the semidirect product
    to take, so that's how it must be. And is the
    normal divisor in the semi-direct product.


  • Gerd Fischer: Analytical Geometry. Vieweg, 1978. ISBN 3-528-17235-5
  • Max Köcher, Aloys Krieg: level geometry. 3rd edition, Springer, Berlin 2007. ISBN 978-3540493273 (p. 102ff deals with the movements of the plane)