Vectorial rotation

See also: Rotation

That is to say E a vector Space Euclidean directed real of finished size N . A vectorial rotation of E is an element of the orthogonal special group $SO\; (E)$. If one chooses a direct orthonormée base of E, its matrix in this base is orthogonal direct.

Plane vectorial rotation

Matric writing

Dans the plan, a vectorial rotation is simply defined by its angle $\backslash \; varphi\; \backslash ,$. Its matrix in a direct orthonormée base is:

In other words, a vector V of components $\backslash \; left;\; there\; \backslash \; right$ for transformed the V' vector of components $\backslash \; left;\; y\text{'}\; \backslash \; right$ that one can calculate with the matric equality:

i.e. one a: $x\text{'}\; =\; X\; \backslash \; cos\; \backslash \; varphi\; -\; there\; \backslash \; sin\; \backslash \; varphi\; \backslash ,$ et
$y\text{'}\; =\; X\; \backslash \; sin\; \backslash \; varphi\; +\; there\; \backslash \; cos\; \backslash \; varphi\; \backslash ,$       

Complex writing

Remark : this can be close to the following formula, written with complex numbers:
$x\text{'}+\; I\; \backslash \; y\text{'}\; =\; (X\; +\; I\; \backslash \; there)\; (\backslash \; cos\; \backslash \; varphi\; +\; I\; \backslash \; sin\; \backslash \; varphi)$ou still: $z\text{'}\; =\; x\text{'}+\; I\; \backslash \; y\text{'}\; =\; (X\; +\; I\; \backslash \; there)\; \backslash \; cdot\; e^\; \{\backslash \; I\; \backslash \; varphi\}\; =\; Z\; \backslash \; cdot\; e^\; \{\backslash \; I\; \backslash \; varphi\}\; \backslash ,$

Direction of rotation

When $\backslash \; varphi$ lies between $0$ and $\backslash \; pi$ and if the plan is directed in a usual way, rotation is made in the trigonometrical or opposite direction needles of a watch. It is said that rotation is sinistral. If $\backslash \; varphi$ lies between $-\; \backslash \; pi$ and $0$, rotation is done in the direction of the needles of a watch. It is known as dextral.

Composition

The made up one of two vectorial rotations is a vectorial rotation whose angle is the sum of the angles of two rotations, which one translates by saying that the group of vectorial rotations is isomorphous with the group $(\backslash \; mathbb\; R/2\; \backslash \; pi\; \backslash \; mathbb\; Z,\; +)$.

Rotations and angles

In the axiomatic construction of the geometry, it is the definition of plane rotations which makes it possible to define the concept of angle (see the article Angle ).

Vectorial rotation in the space of dimension 3

Matric writing

In the space of dimension 3, a vectorial rotation is defined by its axis $\backslash \; vec\; N$ directed whose vectors are invariants by this vectorial rotation and its angle $\backslash \; varphi\; \backslash ,$, that of the plane vectorial rotation which relates to the orthogonal plan with the axis.

We will suppose that the vector $\backslash \; vec\; N$, of coordinates $\backslash \; left;\; n\_y;\; n\_z\; \backslash \; right$ in a direct orthonormée base, is normalized: $\backslash |\backslash \; vec\; NR\; \backslash |\; =\; 1$.

That is to say $\backslash \; vec\; U$ an unspecified vector. Let us note $\backslash \; vec\; V$ its transform by rotation $\backslash \; left\; \backslash \; vec\; NR\; \backslash \; right$.

Simple particular case

Let us start with the study of the particular case where the direct orthonormée base $(\backslash \; vec\; I,\; \backslash \; vec\; J,\; \backslash \; vec\; K)\; \backslash ,$ is such as $\backslash \; vec\; NR\; =\; \backslash \; vec\; k$
Are $\backslash \; mathbf\; \backslash \; pi\; \backslash ,$ the orthogonal vectorial plan with $\backslash \; vec\; N$. Taking into account the particular case, the plan $\backslash \; mathbf\; \backslash \; pi\; \backslash ,$ is the plan generated by the vectors $\backslash \; vec\; i$ and $\backslash \; vec\; j$. The vector $\backslash \; vec\; U$ breaks up into a vector $z\; \backslash \; vec\; K$ colinéaire with $\backslash \; vec\; N$ which is invariant by rotation, and a vector $x\; \backslash \; vec\; I\; +\; there\; \backslash \; vec\; j$ which undergoes a rotation of angle $\backslash \; varphi$ in the plan $\backslash \; mathbf\; \backslash \; Pi$, and one can apply to $x\; \backslash \; vec\; I\; +\; there\; \backslash \; vec\; j$ the formulas established in the case of plane vectorial rotations. One can thus write:

$z\text{'}\; =\; Z\; \backslash ,$            and aussi      comme ci-dessus what can be written in the synthetic form:

General case

If the vector $\backslash \; vec\; N$ has an unspecified orientation compared to the direct orthonormée base $(\backslash \; vec\; I,\; \backslash \; vec\; J,\; \backslash \; vec\; K)\; \backslash ,$ which is used to express the components, the reasoning is more delicate.

As above, let us define the plan $\backslash \; mathbf\; \backslash \; pi\; \backslash ,$, orthogonal with $\backslash \; vec\; N$. The vector $\backslash \; vec\; U$ breaks up into the sum of $(\backslash \; vec\; U\; \backslash \; cdot\; \backslash \; vec\; NR)\; \backslash \; vec\; N$, colinéaire with $\backslash \; vec\; N$ and invariant by rotation, and of $\backslash \; vec\; W\; =\; \backslash \; vec\; U\; -\; (\backslash \; vec\; U\; \backslash \; cdot\; \backslash \; vec\; NR)\; \backslash \; vec\; N$, element of $\backslash \; mathbf\; \backslash \; pi\; \backslash ,$ and which will undergo a rotation in this plan. The directly orthogonal vector with $\backslash \; vec\; W$ in the plan and of the same standard is $\backslash \; vec\; NR\; \backslash \; wedge\; \backslash \; vec\; W$, so that the image of $\backslash \; vec\; W$ in the rotation of angle $\backslash \; varphi$ is $\backslash \; cos\; (\backslash \; varphi)\; \backslash \; vec\; W\; +\; \backslash \; sin\; (\backslash \; varphi)\; \backslash \; vec\; NR\; \backslash \; wedge\; \backslash \; vec\; W$.

Finally, the image of $\backslash \; vec\; U$ by rotation is worth:

$\backslash \; vec\; V\; =\; (\backslash \; vec\; U\; \backslash \; cdot\; \backslash \; vec\; NR)\; \backslash \; vec\; NR\; +\; \backslash \; cos\; (\backslash \; varphi)\; \backslash \; vec\; W\; +\; \backslash \; sin\; (\backslash \; varphi)\; \backslash \; vec\; NR\; \backslash \; wedge\; \backslash \; vec\; W$

and if one replaces $\backslash \; vec\; W$ by his value $\backslash \; vec\; U\; -\; (\backslash \; vec\; U\; \backslash \; cdot\; \backslash \; vec\; NR)\; \backslash \; vec\; N$, one obtains:

$\backslash \; vec\; V\; =\; (\backslash \; vec\; U\; \backslash \; cdot\; \backslash \; vec\; NR)\; \backslash \; vec\; NR\; +\; \backslash \; cos\; (\backslash \; varphi)\; (\backslash \; vec\; U\; -\; (\backslash \; vec\; U\; \backslash \; cdot\; \backslash \; vec\; NR)\; \backslash \; vec\; NR)\; +\; \backslash \; sin\; (\backslash \; varphi)\; \backslash \; vec\; NR\; \backslash \; wedge\; \backslash \; vec\; U$

from where finally the formula of Olinde Rodrigues:

The formula framed above gives the vectorial expression of transformed $\backslash \; vec\; V$ of an unspecified vector $\backslash \; vec\; U$, in rotation $\backslash \; left\; \backslash \; vec\; NR\; \backslash \; right$ of angle $\backslash \; varphi\; \backslash ,$ and of axis $\backslash \; vec\; normalized\; N$ ($n^2\_x+n^2\_y+n^2\_z=1$).

One can have the same result in the following equivalent matric form:

$\backslash \; left\; =\; \backslash \; left\; \backslash \; left$ with:

Remarks

The matrix M is called Matrice of rotation. It is a direct matrix orthogonal, which means that its columns form a direct orthonormée base, or that its transposed matrix is equal to its opposite matrix and that its determinant is worth 1.

Conversely, being given an unspecified matrix of rotation, one easily finds the cosine of the swing angle. Indeed, the trace of the matrix (i.e. the sum of its diagonal elements) is equal to $1\; +\; 2\; \backslash \; cos\; \backslash \; varphi\; \backslash ,$. In addition, it is noticed that:

what makes it possible to quickly find the axis and the sine associated with rotation. Geometrically, $M\; \backslash \; vec\; U$ and $\{\}\; ^t\; M\; \backslash \; vec\; U$ form the two sides of a rhombus whose vector $(M\; -\; \{\}\; ^t\; M)\; \backslash \; vec\; U\; =\; 2\; \backslash \; sin\; (\backslash \; varphi)\; \backslash \; vec\; NR\; \backslash \; wedge\; \backslash \; vec\; U$ is the diagonal, orthogonal with the axis of rotation. It is the rhombus of Olinde Rodrigues.

Composition of two vectorial rotations

The made up one of two vectorial rotations $\backslash \; left\; \backslash \; vec\; N\_1\; \backslash \; right$ and $\backslash \; left\; \backslash \; vec\; N\_2\; \backslash \; right$ of the space of dimension 3 is a vectorial rotation. The characteristics $\backslash \; left\; \backslash \; vec\; N\_3\; \backslash \; right$ of this one are determined quickly starting from $M\_3\; -\; \{\}\; ^t\; M\_3$, where $M\_3$ is the product $M\_2M\_1$ initial matrices of rotation.

One can also call upon the concept of quaternions. Indeed, one can calculate transformed the $\backslash \; vec\; V\; \backslash ,$ of the vector $\backslash \; vec\; U\; \backslash ,$ by using the product of quaternions in the following form:

To compose of rotations then amounts multiplying quaternions.

Related articles

• Rotation S:

• Rotation refines
• Rotation planes
• Rotation in space
• Olinde Rodrigues
• William Rowan Hamilton
• spherical Harmonique
• Quaternion

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