Rotation (mathematics): Difference between revisions
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In [[geometry]] and [[linear algebra]], a '''rotation''' is a [[transformation (mathematics)|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 (mathematics)|translation]], which has no fixed points, and from a [[reflection (mathematics)|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. |
In [[geometry]] and [[linear algebra]], a '''rotation''' is a [[transformation (mathematics)|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 (mathematics)|translation]], which has no fixed points, and from a [[reflection (mathematics)|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. |
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==Two dimensions== |
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[[Image:Rotation4.svg|right|thumb|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 (mathematics)|translation]].]] |
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[[Image:Simx2=rotOK.png|right|thumb|A [[reflection (mathematics)|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.]] |
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It is important to understand the frame of reference when discussing rotations. From one point of view, you may be discussing rotating a vector, keeping the axes fixed. From another point of view, you may be rotating the coordinates, while keeping the vector fixed. |
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In the first point of view, a [[counterclockwise]] rotation of a '''coordinate''' or '''vector''' about the origin, where <math> (x,y) </math> is rotated <math>\theta</math> and we want to know the coordinates after the rotation, <math> (x',y') </math>: |
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:<math> \begin{bmatrix} x' \\ y' \end{bmatrix} = |
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\begin{bmatrix} \cos \theta & -\sin \theta \\ +\sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}. </math> |
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or |
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:<math>x'=x\cos\theta-y\sin\theta\,</math> |
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:<math>y'=+x\sin\theta+y\cos\theta\,</math> |
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A [[counterclockwise]] rotation of the '''plane''' or '''axes''' about the origin, the coordinate in the new plane will be rotated [[clockwise]] in the new coordinates. In this case, if the coordinate in the old plane is <math> (x,y) </math> and the coordinates of the same vector in the new plane is <math> (x',y') </math>, then: |
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:<math> \begin{bmatrix} x' \\ y' \end{bmatrix} = |
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\begin{bmatrix} \cos \theta & +\sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}. </math> |
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or |
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:<math>x'=x\cos\theta+y\sin\theta\,</math> |
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:<math>y'=-x\sin\theta+y\cos\theta\,</math> |
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Then the magnitude of the [[vector (mathematics)|vector]] (''x'', ''y'') is the same as the magnitude of vector (''x''′, ''y''′). |
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===Complex plane=== |
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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 |
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:<math> z = x + iy \,</math> |
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be such a complex number. Its real component is the [[abscissa]] and its imaginary component its [[ordinate]]. |
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Then ''z'' can be rotated counterclockwise by an angle θ by pre-multiplying it with <math> e^{i \theta} </math> (see [[Euler's formula]], §2), viz. |
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{| |
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| <math> e^{i \theta} z \;</math> ||<math> = (\cos \theta + i \sin \theta) (x + i y) \;</math> |
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|- |
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| |
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|<math> = (x \cos \theta + i y \cos \theta + i x \sin \theta - y \sin \theta) \;</math> |
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|- |
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| |
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|<math> = (x \cos \theta - y \sin \theta) + i (x \sin \theta + y \cos \theta) \;</math> |
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|- |
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| |
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|<math> = x' + i y' . \;</math> |
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|} |
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This can be seen to correspond to the rotation described in § 1. |
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Because multiplication of complex numbers is [[commutative]], rotation in 2 dimensions is commutative, unlike in higher dimensions. |
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==Three dimensions== |
==Three dimensions== |
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Revision as of 05:31, 10 September 2007
- For a less mathematical treatment of rotations, see rotation.
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.
Three dimensions
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.
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
An alternative approach to rotation in three dimensions uses 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.
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). See also SO(4).
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.
Relativity
In special relativity a Lorentzian 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.