Editing Rotation (section)

== Mathematics ==
{{Main|Rotation (mathematics)}}
[[File:Rotation illus.svg|right|thumb|Rotation ''([[angular displacement]])'' of a planar figure around a point]]
[[File:Rotation v spin Parent 1.png|thumb|Rotational orbit v spin]]
[[File:AxialTiltObliquity.png|thumb|right|Relations between rotation axis, [[Orbital plane (astronomy)|plane of orbit]] and [[axial tilt]] (for Earth)]]

[[Mathematics|Mathematically]], a rotation is a [[rigid body]] movement which, unlike a [[translation (geometry)|translation]], keeps at least one point fixed. This definition applies to rotations in two dimensions (in a plane), in which exactly one point is kept fixed; and also in three dimensions (in space), in which additional points may be kept fixed (as in rotation around a fixed axis, as infinite line).

All rigid body movements are rotations, translations, or combinations of the two.

A rotation is simply a progressive radial orientation to a common point. That common point lies within the axis of that motion. The axis is perpendicular to the plane of the motion.

If a rotation around a point or axis is followed by a second rotation around the same point/axis, a third rotation results. The ''reverse'' ([[inverse element|''inverse'']]) of a rotation is also a rotation. Thus, the rotations around a point/axis form a [[group (mathematics)|group]]. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, e.g. a translation.

Rotations around the ''x'', ''y'' and ''z'' axes are called ''principal rotations''. Rotation around any axis can be performed by taking a rotation around the ''x'' axis, followed by a rotation around the ''y'' axis, and followed by a rotation around the ''z'' axis. That is to say, any spatial rotation can be decomposed into a combination of principal rotations.

{{See also|curl (mathematics)|cyclic permutation|Euler angles|rigid body|rotation around a fixed axis|rotation group SO(3)|rotation matrix|axis angle|quaternion|isometry}}

=== Fixed axis vs. fixed point ===
The combination of any sequence of rotations of an object in three dimensions about a fixed point is always equivalent to a rotation about an axis (which may be considered to be a rotation in the plane that is perpendicular to that axis). Similarly, the rotation rate of an object in three dimensions at any instant is about some axis, although this axis may be changing over time.

In other than three dimensions, it does not make sense to describe a rotation as being around an axis, since more than one axis through the object may be kept fixed; instead, simple rotations are described as being in a plane. In four or more dimensions, a combination of two or more rotations about a plane is not in general a rotation in a single plane.

=== Axis of 2-dimensional rotations ===
{{further|Rotations in two dimensions}}

2-dimensional rotations, unlike the 3-dimensional ones, possess no axis of rotation, only a point about which the rotation occurs. This is equivalent, for linear transformations, with saying that there is no direction in the plane which is kept unchanged by a 2-dimensional rotation, except, of course, the identity.

The question of the existence of such a direction is the question of existence of an [[eigenvector]] for the matrix ''A'' representing the rotation. Every 2D rotation around the origin through an angle <math>\theta</math> in counterclockwise direction can be quite simply represented by the following [[rotation matrix|matrix]]:

:<math>A =
\begin{bmatrix}
  \cos \theta & -\sin \theta\\
  \sin \theta &  \cos \theta
\end{bmatrix}
</math>

A standard [[eigenvalue]] determination leads to the [[Characteristic polynomial|characteristic equation]]
: <math>\lambda^2 -2 \lambda \cos \theta + 1 = 0,</math>
which has
: <math>\cos \theta \pm i \sin \theta</math>
as its eigenvalues. Therefore, there is no real eigenvalue whenever <math>\cos \theta \neq \pm 1</math>, meaning that no real vector in the plane is kept unchanged by ''A''.

=== Rotation angle and axis in 3 dimensions ===
{{main|Rotation around a fixed axis}}

Knowing that the trace is an invariant, the rotation angle <math>\alpha</math> for a proper orthogonal 3×3 rotation matrix <math>A</math> is found by

:<math>\alpha=\cos^{-1}\left(\frac{A_{11}+A_{22}+A_{33}-1}{2}\right)</math>

Using the principal arc-cosine, this formula gives a rotation angle satisfying <math>0\le\alpha\le 180^\circ</math>. The corresponding rotation axis must be defined to point in a direction that limits the rotation angle to not exceed 180 degrees. (This can always be done because any rotation of more than 180 degrees about an axis <math>m</math> can always be written as a rotation having <math>0\le\alpha\le 180^\circ</math> if the axis is replaced with <math>n=-m</math>.)

Every proper rotation <math>A</math> in 3D space has an axis of rotation, which is defined such that any vector <math>v</math> that is aligned with the rotation axis will not be affected by rotation. Accordingly, <math> A v = v </math>, and the rotation axis therefore corresponds to an eigenvector of the rotation matrix associated with an eigenvalue of 1. As long as the rotation angle <math>\alpha</math> is nonzero (i.e., the rotation is not the identity tensor), there is one and only one such direction. Because A has only real components, there is at least one real eigenvalue, and the remaining two eigenvalues must be complex conjugates of each other (see [[Eigenvalues and eigenvectors#Eigenvalues and the characteristic polynomial]]). Knowing that 1 is an eigenvalue, it follows that the remaining two eigenvalues are complex conjugates of each other, but this does not imply that they are complex—they could be real with double multiplicity. In the degenerate case of a rotation angle <math>\alpha=180^\circ</math>, the remaining two eigenvalues are both equal to −1. In the degenerate case of a zero rotation angle, the rotation matrix is the identity, and all three eigenvalues are 1 (which is the only case for which the rotation axis is arbitrary).

A spectral analysis is not required to find the rotation axis. If <math>n</math> denotes the unit eigenvector aligned with the rotation axis, and if <math>\alpha</math> denotes the rotation angle, then it can be shown that <math>2\sin(\alpha)n=\{A_{32}-A_{23},A_{13}-A_{31},A_{21}-A_{12}\}</math>. Consequently, the expense of an eigenvalue analysis can be avoided by simply normalizing this vector ''if it has a nonzero magnitude.'' On the other hand, if this vector has a zero magnitude, it means that <math>\sin(\alpha)=0</math>. In other words, this vector will be zero if and only if the rotation angle is 0 or 180 degrees, and the rotation axis may be assigned in this case by normalizing any column of <math>A+I</math> that has a nonzero magnitude.<ref>Brannon, R.M., [http://iopscience.iop.org/book/978-0-7503-1454-1 "Rotation, Reflection, and Frame Change"], 2018</ref>

This discussion applies to a proper rotation, and hence <math>\det A = 1</math>. Any improper orthogonal 3x3 matrix <math>B</math> may be written as <math>B=-A</math>, in which <math>A</math> is proper orthogonal. That is, any improper orthogonal 3x3 matrix may be decomposed as a proper rotation (from which an axis of rotation can be found as described above) followed by an inversion (multiplication by −1). It follows that the rotation axis of <math>A</math> is also the eigenvector of <math>B</math> corresponding to an eigenvalue of −1.

===Rotation plane===
{{main|Rotation plane}}

As much as every tridimensional rotation has a rotation axis, also every tridimensional rotation has a plane, which is perpendicular to the rotation axis, and which is left invariant by the rotation. The rotation, restricted to this plane, is an ordinary 2D rotation.

The proof proceeds similarly to the above discussion. First, suppose that all eigenvalues of the 3D rotation matrix ''A'' are real. This means that there is an orthogonal basis, made by the corresponding eigenvectors (which are necessarily orthogonal), over which the effect of the rotation matrix is just stretching it. If we write ''A'' in this basis, it is diagonal; but a diagonal orthogonal matrix is made of just +1s and −1s in the diagonal entries. Therefore, we do not have a proper rotation, but either the identity or the result of a sequence of reflections.

It follows, then, that a proper rotation has some complex eigenvalue. Let ''v'' be the corresponding eigenvector. Then, as we showed in the previous topic, <math> \bar{v} </math> is also an eigenvector, and <math> v + \bar{v} </math> and <math> i(v - \bar{v}) </math> are such that their scalar product vanishes:

:<math> i (v^\text{T} + \bar{v}^\text{T})(v - \bar{v}) = i (v^\text{T} v - \bar{v}^\text{T} \bar{v} + \bar{v}^\text{T} v - v^\text{T} \bar{v} ) = 0 </math>

because, since <math> \bar{v}^\text{T} \bar{v} </math> is real, it equals its complex conjugate <math> v^\text{T} v </math>, and <math> \bar{v}^\text{T} v </math> and <math> v^\text{T} \bar{v} </math> are both representations of the same scalar product between <math> v </math> and <math> \bar{v} </math>.

This means <math> v + \bar{v} </math> and <math> i(v - \bar{v}) </math> are orthogonal vectors. Also, they are both real vectors by construction. These vectors span the same subspace as <math> v </math> and <math> \bar{v} </math>, which is an invariant subspace under the application of ''A''. Therefore, they span an invariant plane.

This plane is orthogonal to the invariant axis, which corresponds to the remaining eigenvector of ''A'', with eigenvalue 1, because of the orthogonality of the eigenvectors of ''A''.

=== Rotation of vectors ===
A vector is said to be rotating if it changes its orientation. This effect is generally only accompanied when its rate of change vector has non-zero perpendicular component to the original vector. This can be shown to be the case by considering a vector <math> \vec A </math> which is parameterized by some variable <math display="inline"> t </math> for which:

<math>{d|\vec A|^2 \over dt}={d(\vec A\cdot \vec A) \over dt} \Rightarrow   {d|\vec A| \over dt}={d\vec A  \over dt}\cdot \hat{A}</math>

Which also gives a relation of rate of change of unit vector by taking <math> \vec A </math>, to be such a vector: <math display="block"> {d\hat A \over dt}\cdot \hat A = 0 </math>showing that <math display="inline"> {d\hat A \over dt} </math> vector is perpendicular to the vector, <math> \vec A </math>.<ref>{{Cite book |last1=Kumar |first1=N. |title=Generalized motion of rigid body |last2=Kumar |first2=Naveen |date=2004 |publisher=Alpha Science International Ltd |isbn=978-1-84265-160-5 |location=Pangbourne, U.K. |pages=5}}</ref>

From:

<math>{d\vec A \over dt} = {d(|\vec A|\hat A) \over dt} = {d|\vec A| \over dt}\hat{A}+|\vec A|\left({d\hat A \over dt}\right)
</math>,

since the first term is parallel to <math> \vec A </math> and the second perpendicular to it, we can conclude in general that the parallel and perpendicular components of rate of change of a vector independently influence only the magnitude or orientation of the vector respectively. Hence, a rotating vector always has a non-zero perpendicular component of its rate of change vector against the vector itself.

===In higher dimensions===
{{see also|Rotations in 4-dimensional Euclidean space}}

As dimensions increase the number of [[rotation vector]]s increases. Along a [[four dimensional space]] (a [[hypervolume]]), rotations occur along x, y, z, and w axis. An object rotated on a w axis intersects through various [[volumes]], where each [[Intersection (geometry)|intersection]] is equal to a self contained volume at an angle. This gives way to a new axis of rotation in a 4d hypervolume, where a 3D object can be rotated perpendicular to the z axis.<ref>{{Cite journal|url=https://ieeexplore.ieee.org/document/6165243|title=Multitouching the Fourth Dimension|first1=Xiaoqi|last1=Yan|first2=Chi-Wing|last2=Fu|first3=Andrew J.|last3=Hanson|date=September 29, 2012|journal=Computer|volume=45|issue=9|pages=80–88|via=Semantic Scholar|doi=10.1109/MC.2012.77}}</ref><ref>{{Cite journal|url=https://doi.org/10.1007/s12650-015-0319-5|title=A visualization method of four-dimensional polytopes by oval display of parallel hyperplane slices|first=Akira|last=Kageyama|date=August 1, 2016|journal=Journal of Visualization|volume=19|issue=3|pages=417–422|via=Springer Link|doi=10.1007/s12650-015-0319-5|arxiv=1607.01102}}</ref>