The special case of a rotation with an internal axis passing through the body's owncenter of mass is known as aspin (orautorotation).[1] In that case, the surface intersection of the internalspin axis can be called apole; for example,Earth's rotation defines thegeographical poles. A rotation around an axis completely external to the moving body is called arevolution (ororbit), e.g.Earth's orbit around theSun. The ends of the externalaxis of revolution can be called theorbital poles.[1]
Either type of rotation is involved in a corresponding type ofangular velocity (spin angular velocity and orbital angular velocity) andangular momentum (spin angular momentum and orbital angular momentum).
Mathematically, a rotation is arigid body movement which, unlike atranslation, 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. Thereverse (inverse) of a rotation is also a rotation. Thus, the rotations around a point/axis form agroup. 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 thex,y andz axes are calledprincipal rotations. Rotation around any axis can be performed by taking a rotation around thex axis, followed by a rotation around they axis, and followed by a rotation around thez axis. That is to say, any spatial rotation can be decomposed into a combination of principal rotations.
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.
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 aneigenvector for the matrixA representing the rotation. Every 2D rotation around the origin through an angle in counterclockwise direction can be quite simply represented by the followingmatrix:
Knowing that the trace is an invariant, the rotation angle for a proper orthogonal 3×3 rotation matrix is found by
Using the principal arc-cosine, this formula gives a rotation angle satisfying. 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 can always be written as a rotation having if the axis is replaced with.)
Every proper rotation in 3D space has an axis of rotation, which is defined such that any vector that is aligned with the rotation axis will not be affected by rotation. Accordingly,, 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 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 (seeEigenvalues 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, 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 denotes the unit eigenvector aligned with the rotation axis, and if denotes the rotation angle, then it can be shown that. Consequently, the expense of an eigenvalue analysis can be avoided by simply normalizing this vectorif it has a nonzero magnitude. On the other hand, if this vector has a zero magnitude, it means that. 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 that has a nonzero magnitude.[2]
This discussion applies to a proper rotation, and hence. Any improper orthogonal 3x3 matrix may be written as, in which 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 is also the eigenvector of corresponding to an eigenvalue of −1.
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 matrixA 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 writeA 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. Letv be the corresponding eigenvector. Then, as we showed in the previous topic, is also an eigenvector, and and are such that their scalar product vanishes:
because, since is real, it equals its complex conjugate, and and are both representations of the same scalar product between and.
This means and are orthogonal vectors. Also, they are both real vectors by construction. These vectors span the same subspace as and, which is an invariant subspace under the application ofA. Therefore, they span an invariant plane.
This plane is orthogonal to the invariant axis, which corresponds to the remaining eigenvector ofA, with eigenvalue 1, because of the orthogonality of the eigenvectors ofA.
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 which is parameterized by some variable for which:
Which also gives a relation of rate of change of unit vector by taking, to be such a vector:showing that vector is perpendicular to the vector,.[3]
From:
,
since the first term is parallel to 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.
As dimensions increase the number ofrotation vectors increases. Along afour dimensional space (ahypervolume), rotations occur along x, y, z, and w axis. An object rotated on a w axis intersects through variousvolumes, where eachintersection is equal to a self contained volume at an angle. This gives way to a new axis of rotation in a 4d hypervolume, were a 3d object can be rotated perpendicular to the z axis.[4][5]
Theangular velocity vector (anaxial vector) also describes the direction of the axis of rotation. Similarly, the torque is an axial vector.
The physics of therotation around a fixed axis is mathematically described with theaxis–angle representation of rotations. According to theright-hand rule, the direction away from the observer is associated with clockwise rotation and the direction towards the observer with counterclockwise rotation, like ascrew.
The motion on the left, an example of curvilinear translation, cannot be treated as rotation since there is no change in orientation, whereas the right can be treated as rotation.
It is possible forobjects to have periodiccircular trajectories without changing theirorientation. These types of motion are treated undercircular motion instead of rotation, more specifically as a curvilinear translation. Since translation involvesdisplacement ofrigid bodies while preserving theorientation of the body, in the case of curvilinear translation, all the points have the same instantaneous velocity whereas relative motion can only be observed in motions involving rotation.[6]
In rotation, theorientation of the object changes and the change inorientation is independent of the observers whoseframes of reference have constant relative orientation over time. ByEuler's theorem, any change in orientation can be described by rotation about an axis through a chosen reference point.[6] Hence, the distinction between rotation and circular motion can be made by requiring an instantaneous axis for rotation, a line passing throughinstantaneous center of circle and perpendicular to theplane of motion. In the example depicting curvilinear translation, the center of circles for the motion lie on a straight line but it is parallel to the plane of motion and hence does not resolve to an axis of rotation. In contrast, a rotating body will always have its instantaneous axis of zero velocity, perpendicular to the plane of motion.[7]
In modern physical cosmology, thecosmological principle is the notion that the distribution of matter in the universe ishomogeneous andisotropic when viewed on a large enough scale, since the forces are expected to act uniformly throughout the universe and have no preferred direction, and should, therefore, produce no observable irregularities in the large scale structuring over the course of evolution of the matter field that was initially laid down by the Big Bang.
Euler rotations of the Earth. Intrinsic (green), Precession (blue) and Nutation (red)
Euler rotations provide an alternative description of a rotation. It is a composition of three rotations defined as the movement obtained by changing one of theEuler angles while leaving the other two constant. Euler rotations are never expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. They constitute a mixed axes of rotation system, where the first angle moves theline of nodes around the external axisz, the second rotates around theline of nodes and the third one is an intrinsic rotation around an axis fixed in the body that moves.
Stars,planets and similar bodies may spin around on their axes. The rotation rate of planets in the solar system was first measured by tracking visual features.Stellar rotation is measured throughDoppler shift or by tracking active surface features. An example issunspots, which rotate around the Sun at the same velocity as theouter gases that make up the Sun.
Under some circumstances orbiting bodies may lock their spin rotation to their orbital rotation around a larger body. This effect is calledtidal locking; the Moon is tidal-locked to the Earth.
This rotation induces acentrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect ofgravitation the closer one is to theequator.Earth's gravity combines both mass effects such that an object weighs slightly less at the equator than at the poles. Another is that over time the Earth is slightly deformed into anoblate spheroid; a similarequatorial bulge develops for other planets.
Another consequence of the rotation of a planet are the phenomena ofprecession andnutation. Like agyroscope, the overall effect is a slight "wobble" in the movement of the axis of a planet. Currently the tilt of theEarth's axis to its orbital plane (obliquity of the ecliptic) is 23.44 degrees, but this angle changes slowly (over thousands of years). (See alsoPrecession of the equinoxes andPole Star.)
Whilerevolution is often used as a synonym forrotation, in many fields, particularly astronomy and related fields,revolution, often referred to asorbital revolution for clarity, is used when one body moves around another whilerotation is used to mean the movement around an axis. Moons revolve around their planets, planets revolve about their stars (such as the Earth around the Sun); and stars slowly revolve about theirgalaxial centers. The motion of the components ofgalaxies is complex, but it usually includes a rotation component.
Mostplanets in theSolar System, includingEarth, spin in the same direction as they orbit theSun. The exceptions areVenus andUranus. Venus may be thought of as rotating slowly backward (or being "upside down"). Uranus rotates nearly on its side relative to its orbit. Current speculation is that Uranus started off with a typical prograde orientation and was knocked on its side by a large impact early in its history. Thedwarf planetPluto (formerly considered a planet) is anomalous in several ways, including that it also rotates on its side.
Inflight dynamics, the principal rotations described withEuler angles above are known aspitch,roll andyaw. The termrotation is also used in aviation to refer to the upward pitch (nose moves up) of an aircraft, particularly when starting the climb after takeoff.
Principal rotations have the advantage of modelling a number of physical systems such asgimbals, andjoysticks, so are easily visualised, and are a very compact way of storing a rotation. But they are difficult to use in calculations as even simple operations like combining rotations are expensive to do, and suffer from a form ofgimbal lock where the angles cannot be uniquely calculated for certain rotations.
Manyamusement rides provide rotation. AFerris wheel has a horizontal central axis, and parallel axes for each gondola, where the rotation is opposite, by gravity or mechanically. As a result, at any time the orientation of the gondola is upright (not rotated), just translated. The tip of the translation vector describes a circle. Acarousel provides rotation about a vertical axis. Many rides provide a combination of rotations about several axes. InChair-O-Planes the rotation about the vertical axis is provided mechanically, while the rotation about the horizontal axis is due to thecentripetal force. Inroller coaster inversions the rotation about the horizontal axis is one or more full cycles, where inertia keeps people in their seats.
Rotation of a player one or more times around a vertical axis may be calledspin infigure skating,twirling (of the baton or the performer) inbaton twirling, or360,540,720, etc. insnowboarding, etc. Rotation of a player or performer one or more times around a horizontal axis may be called aflip,roll,somersault,heli, etc. ingymnastics,waterskiing, or many other sports, or aone-and-a-half,two-and-a-half,gainer (starting facing away from the water), etc. indiving, etc. A combination of vertical and horizontal rotation (back flip with 360°) is called amöbius inwaterskiing freestyle jumping.
Rotation of a player around a vertical axis, generally between 180 and 360 degrees, may be called aspin move and is used as a deceptive or avoidance manoeuvre, or in an attempt to play, pass, or receive a ball or puck, etc., or to afford a player a view of the goal or other players. It is often seen inhockey,basketball,football of various codes,tennis, etc.
