Movement (math)

Motion is a metric space transformation that preserves the distance between the corresponding points, that is, if $A^{'}$ ${\ displaystyle A '}$ $A '$ and $B^{'}$ ${\ displaystyle B '}$ $B '$ - images of points $A$ ${\ displaystyle A}$ $A$ and $B$ ${\ displaystyle B}$ $B$ then $A'B'=AB$ ${\ displaystyle A'B '= AB}$ ${\ displaystyle A'B '= AB}$ . In other words, movement is an isometry of space in itself.

Despite the fact that motion is defined on all metric spaces, this term is more common in Euclidean geometry and related fields. In metric geometry (in particular, in Riemannian geometry ) they often say: isometry of space into itself . In the general case of a metric space (for example, for a non-planar Riemannian manifold ), motions may not always exist.

Sometimes movement is understood to mean a transformation of Euclidean space that preserves orientation. In particular, the axial symmetry of the plane is not considered a movement, and rotation and parallel transport are considered to be movement. Similarly, for general metric spaces, motion is considered an element of the isometry group from the connected component of the identity map .

In Euclidean (or pseudo-Euclidean ) space, motion also automatically preserves angles, so that all scalar products are preserved.

Further in this article, isometrics of only Euclidean point space are considered.

Content

Own and non-proprietary movements

Let be $f\colon E\rightarrow E$ ${\ displaystyle f \ colon E \ rightarrow E}$ - motion of Euclidean point space $E,$ ${\ displaystyle E,}$ but $V$ ${\ displaystyle V}$ - space of free vectors for space $E$ ${\ displaystyle E}$ . Linear operator $Df\colon V\rightarrow V,$ ${\ displaystyle Df \ colon V \ rightarrow V,}$ associated with affine transformation $f,$ ${\ displaystyle f,}$ is an orthogonal operator , and therefore its determinant can be equal to either $one$ ${\ displaystyle 1}$ ( proper orthogonal operator ), or $-1$ ${\ displaystyle -1}$ ( improper orthogonal operator ). In accordance with this, movements are divided into two classes: their own (if $\det Df=1$ ${\ displaystyle \ det Df = 1}$ ) and improper (if $\det Df=-1$ ${\ displaystyle \ det Df = -1}$ ) ^[1] .

Own movements maintain the orientation of space $E,$ ${\ displaystyle E,}$ improper - replace it with the opposite ^[2] . Sometimes proper and improper movements are called displacements and anti- displacements, respectively ^[3] .

Every motion of an n- dimensional Euclidean point space $E$ ${\ displaystyle E}$ can be uniquely determined by indicating an orthonormal frame $(O';e'_{1},\ldots ,e'_{n}),$ ${\ displaystyle (O '; e' _ {1}, \ ldots, e '_ {n}),}$ into which, with a given movement, the previously selected in space passes $E$ ${\ displaystyle E}$ orthonormal frame $(O;e_{1},\ldots ,e_{n}).$ ${\ displaystyle (O; e_ {1}, \ ldots, e_ {n}).}$ Moreover, in the case of own movement, the new frame is oriented in the same way as the original one, and in the case of improper movement, the new frame is oriented in the opposite way. Movements always maintain the distance between points in space $E$ ${\ displaystyle E}$ (that is, they are isometries ), and there are no other isometries other than proper and improper motions ^[4] .

In mechanics, another meaning is implied in the concept of “motion”; in particular, it is always regarded as a continuous process that occurs over a period of time (see mechanical motion ). If, following P. S. Alexandrov , we call continuous motion such a motion of space $E,$ ${\ displaystyle E,}$ which continuously depends on the parameter $t\in [t_{0},t_{1}]$ ${\ displaystyle t \ in [t_ {0}, t_ {1}]}$ (at $n=3$ ${\ displaystyle n = 3}$ in mechanics this corresponds to the movement of an absolutely rigid body ), then the orthonormal frame $(O';e'_{1},\ldots ,e'_{n})$ ${\ displaystyle (O '; e' _ {1}, \ ldots, e '_ {n})}$ can be obtained by continuous movement from an orthonormal frame $(O;e_{1},\ldots ,e_{n})$ ${\ displaystyle (O; e_ {1}, \ ldots, e_ {n})}$ if and only if both frames are oriented identically ^[5] .

Private Isometrics

On the line

Any movement of the line is either a parallel transfer (reduced to the displacement of all points of the line by the same vector lying on the same line), or reflection relative to some point taken on this line. In the first case, the movement is proper, in the second - improper ^[6] .

On the plane

Any movement of the plane refers to one of the following types ^[2] :

Parallel transfer ;
Turn ;
Axial symmetry ( reflection );
Moving symmetry is a superposition of transfer to a vector parallel to the line, and symmetries about this line.

The movements of the first two types are proper, the last two are improper ^[7] .

In three-dimensional space

Any movement of three-dimensional space refers to one of the following types ^[2] :

Parallel transfer;
Turn;
A helical movement is a superposition of rotation relative to a certain line and transfer to a vector parallel to this line;
Mirror symmetry (reflection) relative to the plane ;
Moving symmetry is a superposition of transfer to a vector parallel to the plane and symmetries relative to this plane;
Mirror rotation is a superposition of rotation around a certain line and reflection relative to a plane perpendicular to the axis of rotation.

The motions of the first three types exhaust the class of proper motions of three-dimensional space ( Challe's theorem ), and the motions of the last three types are improper ^[7] .

In n-dimensional space

Superposition of two reflections relative to non-parallel axes gives a rotation

Superposition of two reflections with respect to parallel axes gives parallel transfer

AT $n$ ${\ displaystyle n}$ -dimensional space of motion reduces to orthogonal transformations , parallel transfers and superpositions of both.

In turn, orthogonal transformations can be represented as superpositions of (proper) rotations and mirror reflections (i.e., symmetries with respect to hyperplanes ).

Movements as superpositions of symmetries

Any isometry in $n$ ${\ displaystyle n}$ -dimensional Euclidean space can be represented as a superposition of no more than n + 1 mirror reflections ^[8] .

So, parallel transfer and rotation are superpositions of two reflections, sliding reflection and mirror rotation - three, screw movement - four.

General Isometric Properties

The superposition of isometries is also an isometry ^[9] .
Isometries of the Euclidean space E with respect to the superposition operation form the group Iso ( E ) , which is a Lie group .
Isometry is a special case of the affine transformation (so Iso ( E ) is a subgroup of another Lie group — the affine group Aff ( E ) of the space E ) ^[10] .
The group Iso ( E ) consists of two connected components : the set Iso ₊ ( E ) of proper motions (which is itself a Lie group) and the set Iso _- ( E ) of improper motions; each of these components is linearly connected ^[3] .
Isometry, being an affine transformation , always takes a segment again to a segment.

Notes

↑ Kostrikin and Manin, 1986 , p. 201-204.
↑ ¹ ² ³ Egorov I.P. Movement // Mathematical Encyclopedia. T. 2 / Ch. ed. I.M. Vinogradov . - M .: Soviet Encyclopedia , 1979. - 1104 stb. - Stb. 20-22.
↑ ¹ ² Berger, 1984 , p. 249.
↑ Alexandrov, 1968 , p. 259-262.
↑ Alexandrov, 1968 , p. 210, 214.
↑ Alexandrov, 1968 , p. 284.
↑ ¹ ² Kostrikin and Manin, 1986 , p. 204.
↑ Berger, 1984 , p. 255.
↑ Alexandrov, 1968 , p. 267.
↑ Kostrikin and Manin, 1986 , p. 202.

Literature

Aleksandrov P. S. Lectures on analytic geometry. - M .: Nauka , 1968 .-- 912 p.
Berger M. Geometry. T. 1. - M .: Mir , 1984. - 560 p.
Kostrikin A.I. , Manin Yu. I. Linear algebra and geometry. 2nd ed. - M .: Nauka , 1986 .-- 304 p.

[_0e9ae2be1ff19dcb-1] Kostrikin and Manin, 1986 , p. 201-204.

[Egorov-2] ¹ ² ³ Egorov I.P. Movement // Mathematical Encyclopedia. T. 2 / Ch. ed. I.M. Vinogradov . - M .: Soviet Encyclopedia , 1979. - 1104 stb. - Stb. 20-22.

[_f46ffef2815bcefe-3] ¹ ² Berger, 1984 , p. 249.

[_a0f1fe005cfcb263-4] Alexandrov, 1968 , p. 259-262.

[_163b8de772b2448b-5] Alexandrov, 1968 , p. 210, 214.

[_0210191ce8136015-6] Alexandrov, 1968 , p. 284.

[_a1c3f428c2294c86-7] ¹ ² Kostrikin and Manin, 1986 , p. 204.

[_f46ffdf2815bcd21-8] Berger, 1984 , p. 255.

[_0210131ce81355d8-9] Alexandrov, 1968 , p. 267.

[_a1c3f428c2294c80-10] Kostrikin and Manin, 1986 , p. 202.