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

→ Main article : Congruence mapping

Actual movements of the plane are

• a parallel shift
• a rotation around a point on the plane
  • a point reflection as a special case of a rotation by 180 °

Improper movements are

• an axis mirroring
• a glide mirror consisting of an axis mirroring followed by a translation along the axis.

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)