NationMaster - Encyclopedia: Coordinate rotation

In linear algebra and geometry, a coordinate rotation is a type of transformation from one system of coordinates to another system of coordinates such that distance between any two points remains invariant under the transformation. In other words, a rotation is a type of isometry – note however that there are isometries other than rotations, such as translations, reflections, and glide reflections. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... In geometry, coordinate rotations and reflections are two kinds of isometry which are related to each to other. ... Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ...

1 Two dimensions
- 1.1 Complex plane
2 Three dimensions
- 2.1 Quaternions
3 Generalizations
- 3.1 Orthogonal matrices
- 3.2 Relativity
4 See also

Two dimensions

In two dimensions, a counterclockwise rotation of the plane about the origin, where $(x, y)$ is mapped to $(x', y')$ , is given by the same formulas as a coordinate transformation with a clockwise rotation of the coordinate axes, resulting in a change of coordinates $(x, y)$ into $(x', y')$ :

$begin{bmatrix} x' y' end{bmatrix} = begin{bmatrix} cos theta & -sin theta sin theta & cos theta end{bmatrix} begin{bmatrix} x y end{bmatrix}.$

Be aware of sign of $θ$ if it is important to you. In case you measure angle $θ$ from axis $x$ to $x'$ with positive sign in counterclockwise direction and the both systems $x, y$ and $x', y'$ are lef-handed the rotation matrix has form:

$begin{bmatrix} x' y' end{bmatrix} = begin{bmatrix} cos theta & sin theta -sin theta & cos theta end{bmatrix} begin{bmatrix} x y end{bmatrix}.$

The sign of $sinθ$ depends on how the angle is measured, from axis $x$ to $x'$ or from axis $x'$ to $x$ , if measured angle is positive in clockwise or counterclockwise direction and also if the systems are left-hended or rigt-handed.

Therefor it is sometimes important to find which of these rotation matrix fit to your situation.

In other words

$x'=xcostheta-ysintheta,,$

$y'=xsintheta+ycostheta.,$

Then the magnitude of the vector (x, y) is the same as the magnitude of vector (x′, y′).

Proof. The magnitude of the original vector is

$| (x,y) | = sqrt{x^2 + y^2}$

and the magnitude of the rotated vector is

$| (x',y') | = sqrt{ (x cos theta - y sin theta)^2 + (x sin theta + y cos theta)^2 }$

Expand the squared binomials,

$| (x',y') | = sqrt{ begin{matrix} ; x^2 cos^2 theta - 2xy costheta sintheta + y^2 sin^2 theta + x^2 sin^2 theta + 2xy sintheta costheta + y^2 cos^2 theta end{matrix} }$

$= sqrt{ x^2 cos^2 theta + x^2 sin^2 theta + y^2 sin^2 theta + y^2 cos^2 theta}$

$= sqrt{ x^2 (cos^2 theta + sin^2 theta) + y^2 (sin^2 theta + cos^2 theta) }$

$= sqrt{ x^2 + y^2 }$

Which is the same as the original magnitude.

Complex plane

A complex number can be seen as a two-dimensional vector in the complex plane, with its tail at the origin and its head given by the complex number. Let

$z = x + iy ,$

be such a complex number. Its real component is the abscissa and its imaginary component its ordinate. Abscissa means the x coordinate on an (x, y) graph; the input of a mathematical function against which the output is plotted. ... Ordinate means the y coordinate on an (x, y) graph; the plotted output of a mathematical function. ...

Then z can be rotated counterclockwise by an angle θ by pre-multiplying it with $e i θ$ (see Euler's formula, §2), viz. Eulers formula, named after Leonhard Euler (pronounced oiler), is a mathematical formula in the subfield of complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

$e^{i theta} z ;$	$= (cos theta + i sin theta) (x + i y) ;$
	$= (x cos theta + i y cos theta + i x sin theta - y sin theta) ;$
	$= (x cos theta - y sin theta) + i (x sin theta + y cos theta) ;$
	$= x' + i y' . ;$

This can be seen to correspond to the rotation described in § 1.

Three dimensions

In ordinary three dimensional space, a coordinate rotation can be described by means of Euler angles. It can also be described by means of quaternions (see below), an approach which is similar to the use of vector calculus. Euler angles are the classical way of representing rotations in 3-dimensional Euclidean space, named after Leonhard Euler. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ...

Another way is to multiply by a matrix M, which will rotate space by an angle $θ$ around a unit vector v=(x,y,z), or, alternatively, provides the formulas for converting coordinates if the coordinate axes rotate in opposite direction:

$M(mathbf{v},theta) = begin{bmatrix} cos theta + (1 - cos theta) x^2 & (1 - cos theta) x y - (sin theta) z & (1 - cos theta) x z + (sin theta) y (1 - cos theta) y x + (sin theta) z & cos theta + (1 - cos theta) y^2 & (1 - cos theta) y z - (sin theta) x (1 - cos theta) z x - (sin theta) y & (1 - cos theta) z y + (sin theta) x & cos theta + (1 - cos theta) z^2 end{bmatrix}$

(Compare the equivalent matrix representation in terms of the quaternion components.) The Wikipedia article on quaternions describes the history and purely mathematical properties of the algebra of quaternions. ...

Derivation. This matrix is derived from the following vector algebraic equation (see dot product, cross product, and matrix multiplication): In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ... In mathematics, the cross product is a binary operation on vectors in a three dimensional vector space. ... This article gives an overview of the various ways to multiply matrices. ...

$mathbf{u'} = (cos theta) mathbf{u} + (1 - cos theta) mathbf{v} (mathbf{v} cdot mathbf{u}) + sin theta (mathbf{v} times mathbf{u}), qquad qquad (1)$

which in turn is derived from

Here

$mathbf{u_|} = mathbf{v} ( mathbf{v} cdot mathbf{u}) ,$

$mathbf{u_perp} = mathbf{u} - mathbf{u_|} ,$

$mathbf{v} times mathbf{u_{perp}} = mathbf{v} times mathbf{u} ,$

which shows that u is resolved (see Gram-Schmidt process) into a parallel and a perpendicular component (to v). The parallel component does not rotate, only the perpendicular component does rotate. This rotation is similar to a two dimensional rotation, except that instead of x and y axes, there are $mathbf{u_perp}$ and $mathbf{v} times mathbf{u_perp}$ axes, both of which are perpendicular to v. In mathematics and numerical analysis, the Gram-Schmidt process of linear algebra is a method of orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn. ...

Quaternions

Main article: Quaternions and spatial rotation The Wikipedia article on quaternions describes the history and purely mathematical properties of the algebra of 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. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...

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). This article gives an overview of the various ways to multiply matrices. ... In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ...

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). See also SO(4). In linear algebra, an orthogonal matrix is a square matrix G whose transpose is its inverse, i. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... SO(4) is the symbol used in mathematics for the group of rotations about a fixed point in four-dimensional Euclidean space (for short, the 4D rotation group). ...

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. In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... In mathematics, the unitary group of degree n, denoted U(n), is the group of nÃ—n unitary matrices with complex entries, with the group operation that of matrix multiplication. ... In mathematics, the special unitary group of degree is the group of by unitary matrices with determinant and entries from the field of complex numbers, with the group operation that of matrix multiplication. ... Fig. ... The terms spin and SPIN have several meanings, including those primarily discussed as spinning: For spin in sub-atomic physics, see spin (physics) For the stalled aircraft maneuver or any of several forms of loss of control in aircraft, see spin (flight) For the periodical, see Spin Magazine For the...

Relativity

In special relativity a Lorenzian coordinate rotation which rotates the time axis is called a boost, and, instead of spatial distance, the interval between any two points remains invariant. Lorentzian coordinate rotations which do not rotate the time axis are three dimensional spatial rotations. See: Lorentz transformation, Lorentz group. Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ... A Lorentz transformation (LT) is a linear transformation that preserves the spacetime interval between any two events in Minkowski space, while leaving the origin fixed. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ...

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