Rotation in space

In geometry, the rotation is a transformation of space closely connected Euclidean directed dimension three which has a close relationship with the Rotation planes. Intuitively, this transformation makes turn the figures around a line, called axis, and according to a certain angle. It constitutes a Isométrie because the distances are preserved, and a displacement because it preserves moreover the directed trihedrons.

The complete definition of a rotation requires to direct space as well as the axis of rotation.

Definition

Either E Espace refines Euclidean directed dimension three. One considers a line D space, also directed (for example by the choice of a directing vector). Then any orthogonal plan with D has an orientation induced by the orientations of D and space.

The rotation of angle $\ theta$ and axis D is then the transformation which at a point m associates thus the definite point me

belongs to me to the normal plan with D passing by m ;
and, in this plan provided with the induced orientation, is to me the image of m by the Rotation planes of center H , intersection of the right-hand side and the plan, and angle $\ theta$ .

In particular the points of the axis are invariants. When the angle $\ theta$ is null, rotation is the identical application (whatever the selected axis). Except for this case, a rotation does not admit that only one axis, which is the whole of its points invariants. By construction, the orthogonal plans with the axis are overall invariants.

Rotations whose angle admits for measurement (in radians) $\ frac {\ pi} 2, - \ frac {\ pi} 2, \ pi$ are respectively called quarter of direct turn , quarter of indirect turn and reversal of axis D . In this last case, rotation is also a symmetry: orthogonal symmetry compared to the axis.

Properties

Conservation of the distances and the orientations

Rotations are displacements of space closely connected Euclidean, i.e. Isométrie S respecting the orientation. That means that if has, B, C, D are four points of space and a′ , b′ , c′ , d′ the points images,

the distances ab and a′ b′ is equal;
the Angle S (not directed, since the points are in the space of dimension three) ABC and a′ b′ c′ is equal
the Trièdre abcd is direct if and only if the trihedron a′ b′ c′ d′ is.

Reciprocally, a displacement of space is a rotation if and only if it leaves at least a point invariant. Other displacements are screwings, which can be written like made up of a rotation and a translation parallel to its axis, these two transformations being able to be carried out in any order.

Vectorial rotations

The choice of a point origin O confers on space E a structure of vector Space Euclidean directed. With any rotation $r$ corresponds a vectorial Rotation refines $\ vec r$ which known as is associated with the rotation closely connected, and defined by the relation

\ forall (has, B) \ in E^2, \ qquad \ vec {R} (\ overrightarrow {AB}) = \ overrightarrow {R (A) R (B)}

Vectorial rotations can be identified with rotations closely connected leaving fixes the point origin O , i.e. the identity and rotations whose axis passes by O .

Conversely, an application refines associated with a vectorial rotation is a screwing whose axis is of the same direction.

Composition and decomposition

The made up one of two rotations closely connected is a isometry, and even a displacement, but is not in general a rotation. Moreover, the order in which is carried out the compositions is important. Rotations belong to the group (noncommutative) of displacements of space. Moreover they generate this group, i.e. any displacement can be written like product of rotations. More precisely, any displacement can be written like product of two reversals.

The made up one of two of the same rotations axis is a rotation. For this particular case of composition, the angles are added and the order of composition does not import. Thus the whole of of the same rotations centers, by including the identity there, forms a group commutative, isomorphous with the group $(\ mathbb R/2 \ pi \ mathbb Z, +)$ .

Any rotation can be broken up into a product of two reflections (orthogonal symmetries compared to a plan), one of them being able to be arbitrarily selected.

Traditional formulations of rotations

Rotation according to the angles of Euler

See Angles of Euler.

Rotation by the quaternions

See Quaternion.

See too

vectorial Rotation
Rotation planes
Rotation

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