Rotation matrix

A rotation matrix or the rotation matrix is a matrix mathematics describing a rotation in the Euclidean space. Rotation matrices are orthogonal matrices with determinant 1.

The rotation can be an object ( a figure, a body ) move relative to a fixed coordinate system held or the coordinate system itself.

• 6.1 plane R ²
• 6.2 space R ³

Rotation matrix of the plane R ²

In the Euclidean plane, the rotation of a vector is achieved by a fixed origin by the angle, which is defined in the mathematically positive sense (counterclockwise ) by multiplication with the rotation matrix. There are two types of rotations, rotation of the active and the passive rotation. Wherein the active rotation of the vector is rotated by multiplication with the rotation matrix ( the rotation matrix ) counterclockwise:

In passive rotation, the coordinate system is rotated and hence, the vector is rotated in a clockwise direction. The coordinates of the vector in the rotated coordinate system can be found by multiplication by the matrix:

Each rotation around the origin is a linear map. As with any linear mapping is therefore sufficient to determine the total figure to determine the images of the basis vectors of any basis. Is the standard base is selected, the images of the basis vectors are currently the columns of the associated imaging array.

We have

The rotation matrix for rotation about is therefore

Concatenating two rotations about the angle or is again a rotation, and that the angle. Which belongs to the concatenation matrix can be computed by matrix multiplication,

And allows the reading of the addition formulas for the sine and cosine.

Rotation matrices of the space R ³

The elementary rotations are the rotations about the usual Cartesian coordinate axes. The following matrices rotate a point (or vector) by the angle at fixed coordinate axes. In physics rotations of the coordinate system is often used, the signs of all the sine entries need to be swapped in the matrices below. The rotation by a specific angle of a vector in a coordinate system is equivalent to the rotation of the coordinate system by the same angle in the opposite direction ( negative rotation angle).

The matrices are valid for right- and left-hand systems. Rotations with rotation angles are positive in the legal system turns counterclockwise. In the left-handed system is rotated at positive angles clockwise. The direction of rotation is obtained if one looks at the origin, contrary to the positive axis of rotation. In legal systems may also have a right -hand rule will be applied: Displays the thumb of the right hand in the direction of the axis of rotation, thus giving the diffracted remaining fingers, the direction of the rotation angle on. As a result, the sign of the sine of the entries rotation about the y- axis different from the other two matrices.

• Rotation about the axis:

• Rotation about the axis:

• Rotation about the axis:

• Rotation about a line through the origin whose direction and orientation is given by the arbitrary unit vector:

This arbitrary rotation can be achieved by certain coordinate axes so that this matrix can be formulated with these angles, also three successive rotations with the Euler's angles.

Rotation about an arbitrary axis ( with ) by the angle can be written as in:

This can be rewritten as the Grassmann identity for double cross products and the dyadic product:

Here is the unit matrix and are the canonical unit vectors. The term in curly brackets represents the rotation matrix in dar. In component representation to write this as:

Rotation matrices of the space Rn

In the n- dimensional space rotation is not defined by an axis of rotation, but rather by the plane in which the rotation takes place. Clearly this is clear in two dimensions, where the "rotary axis" is only a point. Let and be two mutually orthogonal unit vectors (ie, and ) that span a level accordingly. Let and. Then give the matrix

A rotation about the angle in the at. It was

And defined. The representation follows from the identities

As well as

Own system of rotation matrices

Everyone on and perpendicular vector ( with ) is mapped by onto itself. So is any vector with an eigenvector of with eigenvalue 1 Two eigenvalues ​​of are the eigenvectors, which is the imaginary unit. From these complex eigenvalues ​​and eigenvectors can thus reconstruct the rotational angle and the rotational plane. Furthermore, valid for rotation in a plane:

However, a rotation in the n- dimensional space in (if n is even ), or (if n is odd ) levels take place at the same time possibly even different angles.

General definition

A matrix R using real components is rotation matrix when

And

Rotation matrices are orthogonal matrices with determinant 1.

Properties

Other properties of rotation matrices:

• Square matrix with real components

• (orthogonal ), it follows from the first part of the definition:

• ( Transpose and inverse of R are the same ), it follows from the orthogonality.

• ( Determinant), corresponds to the second part of the definition.

• The orientation of the coordinate system ( right or left system ) is maintained, since positive orientation.

• The axis of rotation is the solution of

• The angle of rotation is obtained using the scalar product:

• The set of all rotation matrices of a space is the rotation group, namely, the special orthogonal group:

• In addition to the algebraic structure of a group has the set of all rotation matrices also a topological structure: the operations multiplication and inversion of rotation matrices are continuously differentiable functions of their parameters, the rotation angles. The SO (n ) forms a differentiable manifold and is thus a Lie group. This has the dimension.

• With the Lie group SO (n ) is a Lie algebra associated a vector space with a bilinear alternating product ( Lie bracket ), where the vector space is closed under the Lie bracket. This vector space is isomorphic to the tangent space at the neutral element of SO (n ) ( neutral element is the identity matrix ), so that in particular applies. The Lie algebra consists of all skew-symmetric matrices and their basis are the so-called generators. The exponential map combines the Lie algebra with the Lie group:

Infinitesimal rotations

Considering rotations of infinitesimally small angle, it is sufficient to develop the functions of the finite angle of rotation up to a first-order (or ). This now enables represent infinitesimal rotations as

The unit matrix and the generator of an infinitesimal rotation is. The generators are the derivatives of the rotation matrix at the site of identity and form the basis of the Lie algebra ( see example below ).

A finite rotation can be generated by successive application of infinitesimal rotations:

The exponential function is identified. The exponential function of the matrices is defined by the number representation as shown in the last step. It can be shown that generatrix must be traceless:

And skew-symmetric are:

With the concept of generators can be the local group structure of SO (n ) can be expressed in the vicinity of the identity map, by the infinitesimal rotations. Because of the relationship of the exponential function is obtained from a multiplication of rotation matrices an addition to their generators. The generators form a vector space of the same dimension as the rotation group; thus there are linearly independent generator of the group.

The generators form the so-called Lie algebra with the Lie product ( commutator ). An algebra has two group structures, the commutative addition and multiplication (Lie product). The commutator of two generators is back in the set of generators ( seclusion ):

The coefficients are constants characteristic of the group. For all double commutators the Jacobi identity holds:

In theoretical physics Lie groups play an important role, eg in quantum mechanics (see angular momentum operator), or elementary particle physics.

Plane R ²

For rotations in are the infinitesimal rotation and its generatrix:

To SO (2) there is only one linear independent generatrix.

A finite rotation can be represented on the exponential function of the rotation angle and the generatrix. This is shown here on another way: The rotation matrix is decomposed into a symmetric and anti -symmetric part and the trigonometric functions are represented by their Taylor series.

With or follows the result known from above:

Space R ³

For rotations in the Cartesian coordinate axes are the infinitesimal rotations and their generators:

For the SO (3) there are three linearly independent generating. Compared with finite rotations swap infinitesimal rotations of each other ( the commutator vanishes to first order in ).

An infinitesimal rotation and their generating around an arbitrary axis ( with ) can be written as

This fact shows that any generating line is always a skew-symmetric matrix.

A finite rotation around an arbitrary axis ( with ) by the angle can be represented as:

The generators, which form the so-called Lie algebra, ie, the commutator (Lie - product ) of two generators is back in the set of generators:

And also for all cyclic permutations of the indices.