Rotation

Rotation of an object in two dimensions around a point O.

In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. A rotation is different from a translation, which has no fixed points, and from a reflection, which "flips" the bodies it is transforming. A rotation and the above-mentioned transformations are isometries; they leave the distance between any two points unchanged after the transformation.

It is important to know the frame of reference when considering rotations, as all rotations are described relative to a particular frame of reference. In general for any orthogonal transformation on a body in a Coordinate system there is an inverse transformation which if applied to the frame of reference results in the body being at the same coordinates. For example in two dimensions rotating a body clockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed.

In two dimensions

A plane rotation around a point followed by another rotation around a different point results in a total motion which is either a rotation (as in this picture), or a translation.

A reflection against an axis followed by a reflection against a second axis not parallel to the first one results in a total motion that is a rotation around the point of intersection of the axes.

Only a single angle is needed to specify a rotation in two dimensions – the angle of rotation. To calculate the rotation two methods can be used, either matrix algebra or complex numbers.

Matrix algebra

To carry out a rotation using matrices the point (x,y) to be rotated is written as a vector, then multiplied by a matrix calculated from the angle, θ, like so:

.

(x',y') are the co-ordinates of the point after rotation, and the formulae for x' and y' can be seen to be

The vectors and have the same magnitude and are separated by an angle θ as expected.

Complex numbers

Points can also be rotated using complex numbers, as the set of all such numbers, the complex plane, is geometrically a two dimensional plane. the point (x,y) on the plane is represented by the complex number

This can be rotated through an angle θ by multiplying it by e^iθ, then expanding the product using Euler's formula as follows:

which gives the same result as before,

Like complex numbers rotations in two dimensions are commutative, unlike in higher dimensions. They have only one degree of freedom, as such rotations are entirely determined by the angle of rotation.^[1]

In three dimensions

The principal axes of rotation in space

In ordinary three-dimensional space, a coordinate rotation can be defined by three Euler angles, or by a single angle of rotation and the direction of a vector about which to rotate. From these we can deduce that rotations in three dimensions have three degrees of freedom, given by the three Euler angles, or by one the rotation angle plus two for the axis of rotation which lies on a unit sphere.

Rotations about the origin are most easily calculated using a 3×3 matrix transformation called a rotation matrix. Rotations about another point can be described by a 4×4 matrix acting on the homogeneous coordinates.

Quaternions

Quaternions provide another way of representing rotations and orientations in three dimensions. They are applied in computer graphics, control theory, signal processing and orbital mechanics. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude. The rationale is that combining many quaternion transformations is more numerically stable than combining many matrix transformations, and quaternions avoid the problem of gimbal lock.

In four dimensions

A general rotation in four dimensions has only one fixed point, the centre of rotation, and no axis of rotation. Instead the rotation has two mutually orthogonal planes of rotation, each of which is fixed in the sense that points in each plane stay within the planes. The rotation has two angles of rotation, one for each plane of rotation, through which points in the planes rotate. If these are ω₁ and ω₂ then all points not in the planes rotate through an angle between ω₁ and ω₂.

If ω₁ = ω₂ the rotation is a double rotation and all points rotate through the same angle so any two orthogonal planes can be taken as the planes of rotation. If one of ω₁ and ω₂ is zero, one plane is fixed and the rotation is simple. If both ω₁ and ω₂ are zero the rotation is the identity rotation.^[2]

Rotations in four dimensions can be represented by 4th order orthogonal matrices, as a generalisation of the rotation matrix. Quaternions can also be generalised into four dimensions, as even Multivectors of the four dimensional Geometric algebra. A third approach, which only works in four dimensions, is to use a pair of unit quaternions.

Rotations in four dimensions have six degrees of freedom, most easily seen when two unit quaternions are used, as each has three degrees of freedom (they lie on the surface of a 3-sphere) and 2 × 3 = 6.

Relativity

One application of this is special relativity, as it can be considered to operate in a four dimensional space, spacetime, spanned by three space dimensions and one of time. In special relativity this space is linear and the four dimensional rotations, called Lorentz transformations, have practical physical interpretations.

If a simple rotation is only in the three space dimensions, i.e. about a plane that is entirely in space, then this rotation is the same as a spatial rotation in three dimensions. But a simple rotation about a plane spanned by a space dimension and a time dimension is a "boost", a transformation between two different reference frames, which together with other properties of spacetime determines the relativistic relationship between the frames. The set of these rotations forms the Lorentz group.^[3]

Generalizations

Orthogonal matrices

The set of all matrices M(v,θ) described above together with the operation of matrix multiplication is called rotation group: SO(3).

More generally, coordinate rotations in any dimension are represented by orthogonal matrices. The set of all orthogonal matrices of the n-th dimension which describe proper rotations (determinant = +1), together with the operation of matrix multiplication, forms the special orthogonal group: SO(n).

Orthogonal matrices have real elements. The analogous complex-valued matrices are the unitary matrices. The set of all unitary matrices in a given dimension n forms a unitary group of degree n, U(n); and the subgroup of U(n) representing proper rotations forms a special unitary group of degree n, SU(n). The elements of SU(2) are used in quantum mechanics to rotate spin.

See also

Footnotes

References

• Hestenes, David (1999). New Foundations for Classical Mechanics. Dordrecht: Kluwer Academic Publishers. ISBN 0-7923-5514-8. 
• Lounesto, Pertti (2001). Clifford algebras and spinors. Cambridge: Cambridge University Press. ISBN 978-0-521-00551-7. 

External links

• The Mathematics behind rotating and moving observer, explanation on how matrix algebra is used to render 3D scenery viewed by a moving or rotating observer into 2D screen.