# Movement (math)

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.

## definition

A mapping of the -dimensional Euclidean space in itself is called motion if it holds for two points and in${\ displaystyle f \ colon E \ to E}$${\ displaystyle n}$ ${\ displaystyle E}$${\ displaystyle P}$${\ displaystyle Q}$${\ displaystyle E}$

${\ displaystyle d (f (P), f (Q)) \, = \, d (P, Q).}$

Here the Euclidean distance between the points and , i.e. the length of the line or the vector . ${\ displaystyle d (P, Q)}$${\ displaystyle P}$${\ displaystyle Q}$${\ displaystyle {\ overline {PQ}}}$${\ displaystyle {\ overrightarrow {PQ}}}$

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 . ${\ displaystyle n}$${\ displaystyle E}$ ${\ displaystyle V}$

If there is a movement, there is an orthogonal mapping (linear isometry) , so that for all points and : ${\ displaystyle f \ colon E \ to E}$${\ displaystyle {\ vec {f}} \ colon V \ to V}$${\ displaystyle P}$${\ displaystyle Q}$

${\ displaystyle f (Q) = f (P) + {\ vec {f}} ({\ overrightarrow {PQ}})}$

If one chooses an origin , then the following applies to the position vectors of a point and its image point${\ displaystyle O}$${\ displaystyle P}$${\ displaystyle f (P)}$

${\ displaystyle {\ overrightarrow {Of (P)}} = {\ vec {f}} ({\ overrightarrow {OP}}) + {\ overrightarrow {Of (O)}}.}$

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

${\ displaystyle {\ vec {v}} \ mapsto {\ vec {f}} ({\ vec {v}})}$

and translation

${\ displaystyle {\ vec {v}} \ mapsto {\ vec {v}} + {\ overrightarrow {Of (O)}}.}$

## 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 ${\ displaystyle n}$${\ displaystyle E}$ ${\ displaystyle O = X_ {0}, \, X_ {1}, \, \ ldots, \, X_ {n}}$ ${\ displaystyle O}$${\ displaystyle e_ {1} = {\ overrightarrow {OX_ {1}}}, \, \ ldots, \, e_ {n} = {\ overrightarrow {OX_ {n}}}}$${\ displaystyle V}$${\ displaystyle n \ times n}$

${\ displaystyle A = {\ begin {pmatrix} a_ {11} & \ dots & a_ {1n} \\\ vdots && \ vdots \\ a_ {n1} & \ dots & a_ {nn} \ end {pmatrix}}}$

and a translation vector

${\ displaystyle b = {\ begin {pmatrix} b_ {1} \\\ vdots \\ b_ {n} \ end {pmatrix}}}$

describe:

${\ displaystyle y = A \ cdot x + b = {\ begin {pmatrix} a_ {11} & \ dots & a_ {1n} \\\ vdots && \ vdots \\ a_ {n1} & \ dots & a_ {nn} \ end {pmatrix}} \ cdot {\ begin {pmatrix} x_ {1} \\\ vdots \\ x_ {n} \ end {pmatrix}} + {\ begin {pmatrix} b_ {1} \\\ vdots \\ b_ {n} \ end {pmatrix}}}$

Here are

${\ displaystyle x = {\ begin {pmatrix} x_ {1} \\\ vdots \\ x_ {n} \ end {pmatrix}} \ in \ mathbb {R} ^ {n}}$ and ${\ displaystyle y = {\ begin {pmatrix} y_ {1} \\\ vdots \\ y_ {n} \ end {pmatrix}} \ in \ mathbb {R} ^ {n}}$

the coordinate vectors of the position vectors and ${\ displaystyle {\ overrightarrow {OP}} = \ textstyle \ sum \ limits _ {i = 1} ^ {n} x_ {i} e_ {i}}$${\ displaystyle {\ overrightarrow {Of (P)}} = \ textstyle \ sum \ limits _ {i = 1} ^ {n} y_ {i} e_ {i}.}$

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. ${\ displaystyle f}$${\ displaystyle A}$ ${\ displaystyle \ det (A) = 1}$

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

${\ displaystyle (Ax) + b = A (x + c)}$

With ${\ displaystyle c: = A ^ {- 1} b.}$

## 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: ${\ displaystyle \ mathrm {E} (n)}$${\ displaystyle \ mathrm {ISO} (n)}$${\ displaystyle \ mathrm {E} (n)}$${\ displaystyle \ mathrm {E} ^ {+} (n)}$${\ displaystyle \ mathrm {SE} (n)}$ ${\ displaystyle \ mathrm {O} (n) \ ltimes \ mathbb {R} ^ {n}}$${\ displaystyle \ mathrm {SO} (n) \ ltimes \ mathbb {R} ^ {n}}$ ${\ displaystyle \ mathrm {O} (n)}$${\ displaystyle \ mathrm {SO} (n)}$${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle x \ mapsto Ax + a}$${\ displaystyle y \ mapsto By + b}$

${\ displaystyle B (Ax + a) + b = (BA) x + (Ba + b).}$

Both groups are Lie groups of dimension

${\ displaystyle \ dim (\ mathrm {SO} (n)) + \ dim (\ mathbb {R} ^ {n}) = {\ frac {n \ cdot (n-1)} {2}} + n = {\ frac {n \ cdot (n + 1)} {2}}.}$

## 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. ${\ displaystyle [t_ {0}; t_ {1}]}$

## Remarks

1. The associative law is always fulfilled when executing one after the other.
2. The figures and the splitting lemma${\ displaystyle u}$${\ displaystyle v}$
${\ displaystyle 0 \ longrightarrow \ mathbb {R} ^ {n} {\ xrightarrow {\ u \}} \, \ mathrm {E} (n) \, {\ xrightarrow {\ v \}} \, \ mathrm { O} (n) \ longrightarrow 1}$
are with than the unit matrix in and so that with applicable ${\ displaystyle u (a): = (a, I)}$${\ displaystyle I}$${\ displaystyle \ mathrm {O} (n)}$${\ displaystyle v (a, B): = B}$${\ displaystyle r (B): = (0, B)}$
${\ displaystyle v {\ bigl (} r (B) {\ bigr)} = v (0, B) = B.}$
According to the multiplication rule   in   ,
where it is carried out first , then what, however, has no influence on the subsequent term replacement , is therefore the one in the specification of the semidirect product${\ displaystyle (b, B) \ circ (a, A) = (Ba + b, BA)}$${\ displaystyle \ mathrm {E} (n)}$${\ displaystyle (a, A) \; \ colon \, \ mathbb {R} ^ {n} \ to \ mathbb {R} ^ {n}}$${\ displaystyle (b, B)}$${\ displaystyle \ theta \; \ colon \, \ mathrm {O} (n) \ to \ operatorname {Aut} (\ mathbb {R} ^ {n}) = \ mathbb {R} ^ {n}}$${\ displaystyle \ mathbb {R} ^ {n} \! \ rtimes \ mathrm {O} (n)}$
${\ displaystyle {\ begin {array} {lrlclclc} \ theta (B) (a) &: = & u ^ {- 1} {\ bigl (} & r (B) & \ circ & u (a) & \ circ & r ( B ^ {- 1}) & {\ bigr)} \\ & = & u ^ {- 1} {\ bigl (} & (0, B) & \ circ & (a, I) & \ circ & (0, B ^ {- 1}) & {\ bigr)} \\ & = & u ^ {- 1} {\ bigl (} & (0, B) & \ circ & (a, B ^ {- 1}) &&& { \ bigr)} \\ & = & u ^ {- 1} {\ bigl (} & (Ba, I) &&&&& {\ bigr)} \\ & = && Ba \\\ end {array}}}$
to take, so that's how it must be. And is the
normal divisor in the semi-direct product.${\ displaystyle (b, B) \ circ (a, A) = (b + \ theta (B) (a), BA)}$
${\ displaystyle \ mathbb {R} ^ {n}}$

## literature

• 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)