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![f \ colon E \ to E](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f00fa197e33d2f97e56b1b7f0ee709a8c9bd8ee)
![E.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
![Q](https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed)
![E.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
![d (f (P), f (Q)) \, = \, d (P, Q).](https://wikimedia.org/api/rest_v1/media/math/render/svg/5273aebb1fa52cd9518475268e624a6f1a47b69f)
Here the Euclidean distance between the points and , i.e. the length of the line or the vector .
![d (P, Q)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3b0da2776d38a8b78a8eea2fbb9a10cf8cce485)
![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
![Q](https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed)
![{\ overline {PQ}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad2cb75b43d5d9406bd87afc2da2bcbc9eb8241)
![{\ overrightarrow {PQ}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/675febeea8e91072fb11994af206714d0bc598a0)
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 .
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
If there is a movement, there is an orthogonal mapping (linear isometry) , so that for all points and :
![f \ colon E \ to E](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f00fa197e33d2f97e56b1b7f0ee709a8c9bd8ee)
![{\ vec {f}} \ colon V \ to V](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f006d8fd7e7d7453154702c8a4630d2571173a8)
![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
![Q](https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed)
![f (Q) = f (P) + {\ vec {f}} ({\ overrightarrow {PQ}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cddb2a3be82505dd28f28f65addc916dca2536b)
If one chooses an origin , then the following applies to the position vectors of a point and its image point![O](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc)
![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
![\ overrightarrow {Of (P)} = {\ vec f} (\ overrightarrow {OP}) + \ overrightarrow {Of (O)}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/20c5b01220ba88b507ba68e8e8e87f9c9f37d81c)
The position vector of the image point is thus obtained from the composition of the orthogonal mapping
![{\ vec v} \ mapsto {\ vec f} ({\ vec v})](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad3deb073109342b1ccbbf3c362c93f8a66e028e)
and translation
![{\ vec v} \ mapsto {\ vec v} + \ overrightarrow {Of (O)}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/e03eadb5126534bc70b8ea85e456b51ab06a002b)
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
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![O](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc)
![e_ {1} = \ overrightarrow {OX_ {1}}, \, \ ldots, \, e_ {n} = \ overrightarrow {OX_ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5301a39f351d0285c06e8faedfe3f44c23e40009)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![n \ times n](https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78)
![A = {\ begin {pmatrix} a _ {{11}} & \ dots & a _ {{1n}} \\\ vdots && \ vdots \\ a _ {{n1}} & \ dots & a _ {{nn}} \ end { pmatrix}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85d1c207e0463720a085ca3ea6db0596bdc63896)
and a translation vector
![b = {\ begin {pmatrix} b_ {1} \\\ vdots \\ b_ {n} \ end {pmatrix}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bd45eca2134eb31612459977db6df267706ce76)
describe:
![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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3be1193a118edcfa21385d3218af821b23782578)
Here are
-
and
the coordinate vectors of the position vectors and ![\ overrightarrow {OP} = \ textstyle \ sum \ limits _ {{i = 1}} ^ {n} x_ {i} e_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d52db42c2d710e89629840d0565d2b9765963cc0)
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.
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![\ det (A) = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ff759e7103c9bd2ba91a8feda18f042e0785b7)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f04db890d9c968d2eb162be95d0c299ed4b332)
With
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:
![{\ mathrm E} (n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/f699fb0fd3801a34a5016afa897a24953fa9aab5)
![{\ mathrm {ISO}} (n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/775e421637cea2035b48e737635e3d5baf409fcd)
![{\ mathrm E} (n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/f699fb0fd3801a34a5016afa897a24953fa9aab5)
![{\ mathrm {E}} ^ {+} (n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/15557dc0344121c151e42f6a640f78bebc74ea50)
![{\ displaystyle \ mathrm {O} (n) \ ltimes \ mathbb {R} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb3ae0c7c64868ba3dd5849ae5cdee5486833291)
![\ mathrm {O} (n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/1471779b64c8868583dcd50e3c6381293f0dd67f)
![\ mathrm {SO} (n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa71842f19b6810b4bfa9eb282e92fbf285094e)
![\ mathbb {R} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d)
![{\ displaystyle x \ mapsto Ax + a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f305988e31615972577bf8e365e1f329865f2371)
![{\ displaystyle y \ mapsto By + b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cef556c15b6373e350373e1650a7f8887262e74f)
![{\ displaystyle B (Ax + a) + b = (BA) x + (Ba + b).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3e685a494a0182461ed9b7224b0035e7891894)
Both groups are Lie groups of dimension
![\ dim ({\ mathrm {SO}} (n)) + \ dim (\ mathbb {R} ^ {n}) = {\ frac {n \ cdot (n-1)} {2}} + n = { \ frac {n \ cdot (n + 1)} {2}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/60b97cd3db57d24b31f8e6c02112ba569ea3b671)
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.
![[t_ {0}; t_ {1}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/90235c64dd6026fde43331337d114f7100a619dc)
Remarks
-
↑ The associative law is always fulfilled when executing one after the other.
-
↑ The figures and the splitting lemma
![u](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8)
![{\ displaystyle 0 \ longrightarrow \ mathbb {R} ^ {n} {\ xrightarrow {\ u \}} \, \ mathrm {E} (n) \, {\ xrightarrow {\ v \}} \, \ mathrm { O} (n) \ longrightarrow 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a419ab6b1950b729c7c0ccf78b6ae51d11572ef)
are with than the unit matrix in and so that with applicable
![{\ displaystyle u (a): = (a, I)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/525c918cc1fb8e7f31343d00523e4044a6a7892d)
![I.](https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f)
![\ mathrm {O} (n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/1471779b64c8868583dcd50e3c6381293f0dd67f)
![{\ displaystyle v (a, B): = B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5bcdf0d55f864083eda6522bc26f50435ed39f8)
![{\ displaystyle r (B): = (0, B)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d53f102c4ba1aed5f287783bbce19cedc64f38)
![{\ displaystyle v {\ bigl (} r (B) {\ bigr)} = v (0, B) = B.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/039f6934584c060d8f42890c683b6eeb84cd645b)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07fb1255eb1b2abbeba06629fe86d45b01eb3fac)
![{\ displaystyle \ mathrm {E} (n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f699fb0fd3801a34a5016afa897a24953fa9aab5)
![{\ displaystyle (a, A) \; \ colon \, \ mathbb {R} ^ {n} \ to \ mathbb {R} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dd02fe30b1640383e9d6f22e2485e382cfd09fd)
![{\ displaystyle (b, B)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68e52506f696c8d2ed0bf558e6c491bea1ca6e3d)
![{\ displaystyle \ theta \; \ colon \, \ mathrm {O} (n) \ to \ operatorname {Aut} (\ mathbb {R} ^ {n}) = \ mathbb {R} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf6d610b64516afe8e24434187f9e53a596cea4d)
![{\ 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c25371a666cd679e41963331b39bb7aa35f6223f)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d1ac539776149ab5afc5d4e571ec5bd0abf0536)
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)