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54:
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3122:. 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.
1885:
2504:
2904:
448:
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
3180:
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
2938:
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
452:
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
467:
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
3479:. 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. The
344:, 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).
3565:
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. A
1558:
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.
1131:
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
1503:
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
3234:
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 the
2249:
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
1136:). 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
590:
3114:, any change in orientation can be described by rotation about an axis through a chosen reference point. Hence, the distinction between rotation and circular motion can be made by requiring an instantaneous axis for rotation, a line passing through
1169:, 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).
1693:
2304:
1571:
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.
2687:
907:
3545:, 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 of
1332:
951:. 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
1566:
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
2602:
663:
3743:
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 in
1936:
1008:
949:
3433:
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 their
710:
350:
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.
3095:
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.
2044:
2005:
748:
3063:
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.
1167:
509:
2648:
1376:. 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
2175:
1685:
1374:
2973:
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.
1966:
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2677:
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2277:
2224:
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1606:
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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 an
1432:
1210:
1129:
786:
497:
1880:{\displaystyle 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}
1481:
1109:
1037:
1400:
3230:
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 the
2499:{\displaystyle {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}}}
2195:
2064:
1542:
1522:
1501:
1452:
1190:
1080:
1060:
969:
806:
2899:{\displaystyle {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)}
2297:
367:. 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.
3037:, the direction away from the observer is associated with clockwise rotation and the direction towards the observer with counterclockwise rotation, like a
2948:
1111:, 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
814:
3318:
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.
1215:
3825:
4186:
3653:
paddles are manufactured with different surface characteristics to allow the player to impart a greater or lesser amount of spin to the ball.
2540:
4137:
4054:
4023:
3915:
3871:
3830:
3337:
Under some circumstances orbiting bodies may lock their spin rotation to their orbital rotation around a larger body. This effect is called
3534:
is also used in aviation to refer to the upward pitch (nose moves up) of an aircraft, particularly when starting the climb after takeoff.
151:
137:
3205:
3126:
353:
If a rotation around a point or axis is followed by a second rotation around the same point/axis, a third rotation results. The
4294:
4151:
3345:
608:
75:
3030:
4114:
3574:
the rotation about the vertical axis is provided mechanically, while the rotation about the horizontal axis is due to the
4165:
4078:
4289:
3820:
3142:
3054:
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1133:
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418:
202:
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118:
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462:
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1893:
90:
1562:
The proof proceeds similarly to the above discussion. First, suppose that all eigenvalues of the 3D rotation matrix
974:
915:
3786:
3591:
3115:
674:
2177:
are orthogonal vectors. Also, they are both real vectors by construction. These vectors span the same subspace as
64:
3388:
3282:
600:
322:
71:
31:
3795:– speculative hypothesis that a physical law relates the motion of the distant stars to the local inertial frame
97:
3579:
3001:(seconds, days, etc.). The time-rate of change of angular frequency is angular acceleration (rad/s), caused by
3582:
the rotation about the horizontal axis is one or more full cycles, where inertia keeps people in their seats.
3570:
provides rotation about a vertical axis. Many rides provide a combination of rotations about several axes. In
179:
or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a
3383:, the overall effect is a slight "wobble" in the movement of the axis of a planet. Currently the tilt of the
585:{\displaystyle A={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}}
3495:
3303:, rotation is a commonly observed phenomenon; it includes both spin (auto-rotation) and orbital revolution.
3084:
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2010:
1971:
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3486:(formerly considered a planet) is anomalous in several ways, including that it also rotates on its side.
3626:
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2607:
398:
axis. That is to say, any spatial rotation can be decomposed into a combination of principal rotations.
4174:
2965:), rotations occur along x, y, z, and w axis. An object rotated on a w axis intersects through various
1334:. Consequently, the expense of an eigenvalue analysis can be avoided by simply normalizing this vector
3517:
3193:
3185:
2136:
1646:
1341:
422:
300:
86:
3964:"A visualization method of four-dimensional polytopes by oval display of parallel hyperplane slices"
3739:
Rotation of a player around a vertical axis, generally between 180 and 360 degrees, may be called a
3688:, etc. Rotation of a player or performer one or more times around a horizontal axis may be called a
4299:
4251:
3814:
3792:
3531:
3294:
3278:
3022:) also describes the direction of the axis of rotation. Similarly, the torque is an axial vector.
1941:
364:
233:
3184:
In particular, for a system which behaves the same regardless of how it is oriented in space, its
2233:
This plane is orthogonal to the invariant axis, which corresponds to the remaining eigenvector of
2101:
1611:
4239:
4227:
4179:
4158:
Rotation, Reflection, and Frame Change: Orthogonal tensors in computational engineering mechanics
3975:
3408:
3201:
3107:
406:
402:
190:
181:
3357:
3083:
instead of rotation, more specifically as a curvilinear translation. Since translation involves
2912:
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2511:
2253:
2200:
2069:
1582:
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4104:
4050:
4040:
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1114:
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3985:
3944:
3803:
3689:
3365:
3319:
3197:
3013:
276:
272:
251:
194:
3728:, etc. A combination of vertical and horizontal rotation (back flip with 360°) is called a
3500:
3775:
3642:
3509:
3434:
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426:
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309:
3537:
Principal rotations have the advantage of modelling a number of physical systems such as
1379:
3247:
and the third one is an intrinsic rotation around an axis fixed in the body that moves.
4263:
4191:
3725:
3669:
3661:
3571:
3558:
3391:) is 23.44 degrees, but this angle changes slowly (over thousands of years). (See also
2180:
2049:
1553:
1527:
1507:
1486:
1437:
1175:
1065:
1045:
954:
791:
213:
42:
1082:
that is aligned with the rotation axis will not be affected by rotation. Accordingly,
111:
4273:
3338:
3323:
3244:
3236:
3226:
Euler rotations of the Earth. Intrinsic (green), Precession (blue) and
Nutation (red)
471:
The question of the existence of such a direction is the question of existence of an
38:
212:
The special case of a rotation with an internal axis passing through the body's own
4258:
4215:
4157:
4125:
3893:
3835:
3685:
3650:
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3562:
3480:
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3018:
2282:
410:
363:) of a rotation is also a rotation. Thus, the rotations around a point/axis form a
264:
186:
172:
3141:, called general plane motion. A simple example of pure rotation is considered in
902:{\displaystyle \alpha =\cos ^{-1}\left({\frac {A_{11}+A_{22}+A_{33}-1}{2}}\right)}
347:
All rigid body movements are rotations, translations, or combinations of the two.
317:
4044:
4013:
449:
at any instant is about some axis, although this axis may be changing over time.
3932:
3733:
3709:
3646:
3546:
3349:
3331:
3173:
3161:. (Although they do appear to change when viewed from a rotating viewpoint: see
2962:
479:
representing the rotation. Every 2D rotation around the origin through an angle
472:
333:
240:. A rotation around an axis completely external to the moving body is called a
53:
912:
Using the principal arc-cosine, this formula gives a rotation angle satisfying
499:
in counterclockwise direction can be quite simply represented by the following
4070:
3989:
3748:
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3274:
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in the reference frame of the Earth which slightly counteracts the effect of
1327:{\displaystyle 2\sin(\alpha )n=\{A_{32}-A_{23},A_{13}-A_{31},A_{21}-A_{12}\}}
3963:
3656:
Rotation of a player one or more times around a vertical axis may be called
3396:
3380:
3300:
3177:
2998:
2990:
1579:
be the corresponding eigenvector. Then, as we showed in the previous topic,
386:. Rotation around any axis can be performed by taking a rotation around the
293:
176:
1062:
in 3D space has an axis of rotation, which is defined such that any vector
4041:"Planar kinematics of a rigid body: Instantaneous center of zero velocity"
3204:
of its
Lagrangian) of a physical system is invariant under rotation, then
1575:
It follows, then, that a proper rotation has some complex eigenvalue. Let
1134:
Eigenvalues and eigenvectors#Eigenvalues and the characteristic polynomial
3752:
3634:
3606:
3567:
3542:
3376:
3327:
3255:
2237:, with eigenvalue 1, because of the orthogonality of the eigenvectors of
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165:
is the circular movement of an object around a central line, known as an
3222:
17:
3948:
3808:
3780:
3602:
3353:
2966:
2508:
Which also gives a relation of rate of change of unit vector by taking
30:
This article is about movement of a physical body. For other uses, see
3549:
where the angles cannot be uniquely calculated for certain rotations.
1338:
On the other hand, if this vector has a zero magnitude, it means that
3756:
3744:
3610:
3538:
3476:
3456:
3438:
3371:
Another consequence of the rotation of a planet are the phenomena of
3315:
3002:
1192:
denotes the unit eigenvector aligned with the rotation axis, and if
715:
as its eigenvalues. Therefore, there is no real eigenvalue whenever
4210:
3980:
3931:
Yan, Xiaoqi; Fu, Chi-Wing; Hanson, Andrew J. (September 29, 2012).
3811:– motion of two objects in contact with each-other without sliding
3499:
3483:
3472:
3464:
3384:
3269:
3221:
3038:
3005:. The ratio of torque to the angular acceleration is given by the
1172:
A spectral analysis is not required to find the rotation axis. If
316:
308:
292:
246:
149:
3789:– instantaneously fixed point on an arbitrarily moving rigid body
3172:
is the notion that the distribution of matter in the universe is
3910:. Pangbourne, U.K.: Alpha Science International Ltd. p. 5.
3311:
750:, meaning that no real vector in the plane is kept unchanged by
3863:
Metaphors & Analogies: Power Tools for
Teaching Any Subject
271:
Either type of rotation is involved in a corresponding type of
3468:
3429:
for clarity, is used when one body moves around another while
255:
47:
4146:
by Sergio
Hannibal Mejia after work by Roger Germundsson and
3421:, in many fields, particularly astronomy and related fields,
2046:
are both representations of the same scalar product between
768:
Knowing that the trace is an invariant, the rotation angle
453:
in a plane is not in general a rotation in a single plane.
3441:
is complex, but it usually includes a rotation component.
3330:, which rotate around the Sun at the same velocity as the
2226:, which is an invariant subspace under the application of
224:). In that case, the surface intersection of the internal
275:(spin angular velocity and orbital angular velocity) and
3817:– a unitless scalar representing the number of rotations
2597:{\displaystyle {d{\hat {A}} \over dt}\cdot {\hat {A}}=0}
1405:
This discussion applies to a proper rotation, and hence
658:{\displaystyle \lambda ^{2}-2\lambda \cos \theta +1=0,}
3772: – Rotation independent of any external reference
3326:
or by tracking active surface features. An example is
2610:
2285:
1212:
denotes the rotation angle, then it can be shown that
524:
279:(spin angular momentum and orbital angular momentum).
4163:
2939:
its rate of change vector against the vector itself.
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3597:Rotation of a ball or other object, usually called
78:. Unsourced material may be challenged and removed.
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3110:have constant relative orientation over time. By
3590:"Spin move" redirects here. For other uses, see
1412:
3724:(starting facing away from the water), etc. in
3467:, spin in the same direction as they orbit the
1931:{\displaystyle {\bar {v}}^{\text{T}}{\bar {v}}}
1003:{\displaystyle 0\leq \alpha \leq 180^{\circ }}
944:{\displaystyle 0\leq \alpha \leq 180^{\circ }}
4046:Engineering Mechanics: Statics & dynamics
3855:
3853:
1687:are such that their scalar product vanishes:
705:{\displaystyle \cos \theta \pm i\sin \theta }
8:
1321:
1243:
788:for a proper orthogonal 3×3 rotation matrix
394:axis, and followed by a rotation around the
189:has an infinite number of possible axes and
2230:. Therefore, they span an invariant plane.
971:can always be written as a rotation having
37:"Rotate" redirects here. For the song, see
4008:Harrison, H.; Nettleton, T. (1997-08-01).
3079:. These types of motion are treated under
2949:Rotations in 4-dimensional Euclidean space
154:A sphere rotating (spinning) about an axis
3979:
3601:, plays a role in many sports, including
3512:, the principal rotations described with
3341:; the Moon is tidal-locked to the Earth.
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138:Learn how and when to remove this message
3894:"Rotation, Reflection, and Frame Change"
3102:of the object changes and the change in
3058:
2279:which is parameterized by some variable
390:axis, followed by a rotation around the
4170:
4010:"Rigid body motion in three dimensions"
3849:
3826:Rotation formalisms in three dimensions
3504:The principal axes of rotation in space
2650:vector is perpendicular to the vector,
758:Rotation angle and axis in 3 dimensions
3106:is independent of the observers whose
2039:{\displaystyle v^{\text{T}}{\bar {v}}}
2000:{\displaystyle {\bar {v}}^{\text{T}}v}
1544:corresponding to an eigenvalue of −1.
743:{\displaystyle \cos \theta \neq \pm 1}
4138:Rotate Points Using Polar Coordinates
4018:. Butterworth-Heinemann. p. 55.
3831:Rotating locomotion in living systems
3029:is mathematically described with the
2953:As dimensions increase the number of
1434:. Any improper orthogonal 3x3 matrix
7:
4081:from the original on 11 October 2013
4003:
4001:
3999:
3933:"Multitouching the Fourth Dimension"
3866:. Stenhouse Publishers. p. 28.
3513:
2909:since the first term is parallel to
1162:{\displaystyle \alpha =180^{\circ }}
76:adding citations to reliable sources
27:Movement of an object around an axis
3668:(of the baton or the performer) in
3133:can be treated as a composition of
2643:{\textstyle {d{\hat {A}} \over dt}}
3962:Kageyama, Akira (August 1, 2016).
3437:. The motion of the components of
3168:In modern physical cosmology, the
3159:invariant under any fixed rotation
25:
4134:at cut-the-knot. cut-the-knot.org
3906:Kumar, N.; Kumar, Naveen (2004).
321:Relations between rotation axis,
305:of a planar figure around a point
4257:
4245:
4233:
4221:
4209:
4197:
4185:
4173:
3908:Generalized motion of rigid body
3243:, the second rotates around the
52:
3783:– large scale rotating air mass
3417:is often used as a synonym for
3033:of rotations. According to the
2170:{\displaystyle i(v-{\bar {v}})}
1680:{\displaystyle i(v-{\bar {v}})}
1369:{\displaystyle \sin(\alpha )=0}
457:Axis of 2-dimensional rotations
63:needs additional citations for
4152:Wolfram Demonstrations Project
4132:When a Triangle is Equilateral
3387:'s axis to its orbital plane (
3116:instantaneous center of circle
2922:
2875:
2855:
2848:
2838:
2828:
2807:
2800:
2790:
2766:
2760:
2750:
2743:
2733:
2729:
2703:
2663:
2623:
2582:
2556:
2521:
2490:
2464:
2434:
2427:
2417:
2407:
2393:
2387:
2372:
2363:
2333:
2325:
2315:
2263:
2210:
2164:
2158:
2143:
2117:
2079:
2030:
1982:
1922:
1904:
1868:
1862:
1828:
1812:
1794:
1768:
1759:
1753:
1738:
1735:
1723:
1700:
1674:
1668:
1653:
1627:
1592:
1402:that has a nonzero magnitude.
1357:
1351:
1336:if it has a nonzero magnitude.
1234:
1228:
1:
4071:"An Oasis, or a Secret Lair?"
4015:Advanced Engineering Dynamics
3951:– via Semantic Scholar.
3734:waterskiing freestyle jumping
3712:, or many other sports, or a
3206:angular momentum is conserved
3157:are currently believed to be
1961:{\displaystyle v^{\text{T}}v}
1010:if the axis is replaced with
4154:. demonstrations.wolfram.com
3821:Rotation around a fixed axis
3368:develops for other planets.
3143:rotation around a fixed axis
3055:Rotation around a fixed axis
3027:rotation around a fixed axis
2126:{\displaystyle v+{\bar {v}}}
1636:{\displaystyle v+{\bar {v}}}
1608:is also an eigenvector, and
764:Rotation around a fixed axis
419:rotation around a fixed axis
4110:Encyclopedia of Mathematics
3393:Precession of the equinoxes
3250:These rotations are called
3181:laid down by the Big Bang.
3163:rotating frame of reference
1524:is also the eigenvector of
599:determination leads to the
463:Rotations in two dimensions
258:. The ends of the external
4316:
4144:Rotation in Two Dimensions
3992:– via Springer Link.
3787:Instant centre of rotation
3592:Spin move (disambiguation)
3589:
3493:
3448:
3406:
3288:
3215:
3048:
2946:
2931:{\displaystyle {\vec {A}}}
2672:{\displaystyle {\vec {A}}}
2530:{\displaystyle {\vec {A}}}
2272:{\displaystyle {\vec {A}}}
2219:{\displaystyle {\bar {v}}}
2088:{\displaystyle {\bar {v}}}
1601:{\displaystyle {\bar {v}}}
1551:
761:
460:
444:Fixed axis vs. fixed point
400:
286:
41:. For the ghost town, see
36:
29:
4148:Understanding 3D Rotation
3990:10.1007/s12650-015-0319-5
3580:roller coaster inversions
3389:obliquity of the ecliptic
3352:the closer one is to the
3239:around the external axis
3118:and perpendicular to the
3031:axis–angle representation
340:movement which, unlike a
32:Rotation (disambiguation)
4039:Hibbeler, R. C. (2007).
3968:Journal of Visualization
3346:centrifugal acceleration
3344:This rotation induces a
1427:{\displaystyle \det A=1}
4075:ESO Picture of the Week
3496:Aircraft principal axes
3425:, often referred to as
3125:More generally, due to
3075:without changing their
2537:, to be such a vector:
1205:{\displaystyle \alpha }
1124:{\displaystyle \alpha }
781:{\displaystyle \alpha }
601:characteristic equation
492:{\displaystyle \theta }
313:Rotational orbit v spin
175:can rotate in either a
4295:Orientation (geometry)
4150:by Roger Germundsson,
3799:Orientation (geometry)
3505:
3334:that make up the Sun.
3286:
3283:camera's long exposure
3227:
3190:rotationally invariant
3170:cosmological principle
3149:Cosmological principle
3064:
2959:four dimensional space
2932:
2900:
2673:
2644:
2598:
2531:
2500:
2293:
2273:
2220:
2191:
2171:
2127:
2089:
2060:
2040:
2001:
1962:
1932:
1881:
1681:
1637:
1602:
1538:
1518:
1497:
1477:
1448:
1428:
1396:
1370:
1328:
1206:
1186:
1163:
1125:
1105:
1076:
1056:
1042:Every proper rotation
1033:
1004:
965:
945:
903:
802:
782:
744:
706:
659:
586:
493:
468:course, the identity.
330:
314:
306:
289:Rotation (mathematics)
155:
3503:
3471:. The exceptions are
3289:Further information:
3273:
3225:
3091:while preserving the
3073:circular trajectories
3062:
2933:
2901:
2674:
2645:
2599:
2532:
2501:
2294:
2274:
2221:
2192:
2172:
2128:
2090:
2061:
2041:
2002:
1963:
1933:
1882:
1682:
1638:
1603:
1539:
1519:
1498:
1478:
1449:
1429:
1397:
1371:
1329:
1207:
1187:
1164:
1126:
1106:
1077:
1057:
1034:
1005:
966:
946:
904:
803:
783:
745:
707:
660:
587:
494:
461:Further information:
370:Rotations around the
320:
312:
296:
153:
4122:Product of Rotations
3860:Wormeli, R. (2009).
3561:provide rotation. A
3322:is measured through
2943:In higher dimensions
2913:
2688:
2654:
2608:
2541:
2512:
2305:
2283:
2254:
2201:
2181:
2137:
2102:
2070:
2050:
2011:
1972:
1942:
1894:
1694:
1647:
1612:
1583:
1528:
1508:
1487:
1476:{\displaystyle B=-A}
1458:
1438:
1409:
1380:
1342:
1216:
1196:
1176:
1140:
1115:
1104:{\displaystyle Av=v}
1086:
1066:
1046:
1032:{\displaystyle n=-m}
1014:
975:
955:
916:
815:
792:
772:
719:
675:
609:
510:
483:
423:rotation group SO(3)
301:angular displacement
72:improve this article
4290:Classical mechanics
4140:, howtoproperly.com
3815:Rotation (quantity)
3445:Retrograde rotation
3108:frames of reference
3067:It is possible for
3025:The physics of the
2957:increases. Along a
2245:Rotation of vectors
1395:{\displaystyle A+I}
384:principal rotations
197:(between arbitrary
4285:Euclidean geometry
4128:. cut-the-knot.org
3949:10.1109/MC.2012.77
3755:of various codes,
3627:billiards and pool
3514:Euler angles above
3506:
3427:orbital revolution
3409:Orbital revolution
3287:
3260:intrinsic rotation
3228:
3202:integral over time
3065:
2928:
2896:
2669:
2640:
2594:
2527:
2496:
2289:
2269:
2216:
2187:
2167:
2123:
2085:
2056:
2036:
1997:
1958:
1928:
1877:
1677:
1633:
1598:
1534:
1514:
1493:
1473:
1454:may be written as
1444:
1424:
1392:
1366:
1324:
1202:
1182:
1159:
1121:
1101:
1072:
1052:
1029:
1000:
961:
941:
899:
798:
778:
740:
702:
655:
582:
576:
489:
407:cyclic permutation
403:curl (mathematics)
336:, a rotation is a
331:
315:
307:
262:can be called the
260:axis of revolution
238:geographical poles
203:rotation around a
201:), in contrast to
191:angles of rotation
182:center of rotation
156:
4056:978-0-13-221509-1
4049:. Prentice-Hall.
4025:978-0-08-052335-4
3917:978-1-84265-160-5
3873:978-1-57110-758-9
3770:Absolute rotation
3576:centripetal force
3451:Retrograde motion
3194:Noether's theorem
3098:In rotation, the
3071:to have periodic
3007:moment of inertia
2987:angular frequency
2983:speed of rotation
2925:
2890:
2878:
2851:
2831:
2820:
2803:
2778:
2763:
2746:
2718:
2706:
2666:
2638:
2626:
2585:
2571:
2559:
2524:
2493:
2479:
2467:
2447:
2430:
2405:
2390:
2375:
2352:
2328:
2266:
2213:
2190:{\displaystyle v}
2161:
2120:
2082:
2059:{\displaystyle v}
2033:
2021:
1991:
1985:
1952:
1925:
1913:
1907:
1865:
1853:
1837:
1831:
1815:
1803:
1797:
1778:
1756:
1732:
1726:
1710:
1671:
1630:
1595:
1537:{\displaystyle B}
1517:{\displaystyle A}
1496:{\displaystyle A}
1447:{\displaystyle B}
1185:{\displaystyle n}
1075:{\displaystyle v}
1055:{\displaystyle A}
964:{\displaystyle m}
893:
801:{\displaystyle A}
163:rotational motion
148:
147:
140:
122:
16:(Redirected from
4307:
4262:
4261:
4250:
4249:
4248:
4238:
4237:
4236:
4226:
4225:
4224:
4214:
4213:
4202:
4201:
4200:
4190:
4189:
4178:
4177:
4169:
4160:, IOP Publishing
4118:
4091:
4090:
4088:
4086:
4067:
4061:
4060:
4036:
4030:
4029:
4005:
3994:
3993:
3983:
3959:
3953:
3952:
3928:
3922:
3921:
3903:
3897:
3890:
3884:
3883:
3881:
3880:
3857:
3804:Point reflection
3793:Mach's principle
3435:galaxial centers
3366:equatorial bulge
3320:Stellar rotation
3295:Earth's rotation
3279:Earth's rotation
3129:, any motion of
3127:Chasles' theorem
3014:angular velocity
2985:is given by the
2955:rotation vectors
2937:
2935:
2934:
2929:
2927:
2926:
2918:
2905:
2903:
2902:
2897:
2895:
2891:
2889:
2881:
2880:
2879:
2871:
2864:
2858:
2853:
2852:
2844:
2841:
2833:
2832:
2824:
2821:
2819:
2811:
2810:
2805:
2804:
2796:
2793:
2784:
2779:
2777:
2769:
2765:
2764:
2756:
2753:
2748:
2747:
2739:
2736:
2724:
2719:
2717:
2709:
2708:
2707:
2699:
2692:
2678:
2676:
2675:
2670:
2668:
2667:
2659:
2649:
2647:
2646:
2641:
2639:
2637:
2629:
2628:
2627:
2619:
2612:
2603:
2601:
2600:
2595:
2587:
2586:
2578:
2572:
2570:
2562:
2561:
2560:
2552:
2545:
2536:
2534:
2533:
2528:
2526:
2525:
2517:
2505:
2503:
2502:
2497:
2495:
2494:
2486:
2480:
2478:
2470:
2469:
2468:
2460:
2453:
2448:
2446:
2438:
2437:
2432:
2431:
2423:
2420:
2411:
2406:
2404:
2396:
2392:
2391:
2383:
2377:
2376:
2368:
2358:
2353:
2351:
2343:
2342:
2341:
2336:
2330:
2329:
2321:
2318:
2309:
2298:
2296:
2295:
2290:
2278:
2276:
2275:
2270:
2268:
2267:
2259:
2225:
2223:
2222:
2217:
2215:
2214:
2206:
2196:
2194:
2193:
2188:
2176:
2174:
2173:
2168:
2163:
2162:
2154:
2132:
2130:
2129:
2124:
2122:
2121:
2113:
2094:
2092:
2091:
2086:
2084:
2083:
2075:
2065:
2063:
2062:
2057:
2045:
2043:
2042:
2037:
2035:
2034:
2026:
2023:
2022:
2019:
2006:
2004:
2003:
1998:
1993:
1992:
1989:
1987:
1986:
1978:
1967:
1965:
1964:
1959:
1954:
1953:
1950:
1937:
1935:
1934:
1929:
1927:
1926:
1918:
1915:
1914:
1911:
1909:
1908:
1900:
1886:
1884:
1883:
1878:
1867:
1866:
1858:
1855:
1854:
1851:
1839:
1838:
1835:
1833:
1832:
1824:
1817:
1816:
1808:
1805:
1804:
1801:
1799:
1798:
1790:
1780:
1779:
1776:
1758:
1757:
1749:
1734:
1733:
1730:
1728:
1727:
1719:
1712:
1711:
1708:
1686:
1684:
1683:
1678:
1673:
1672:
1664:
1642:
1640:
1639:
1634:
1632:
1631:
1623:
1607:
1605:
1604:
1599:
1597:
1596:
1588:
1543:
1541:
1540:
1535:
1523:
1521:
1520:
1515:
1502:
1500:
1499:
1494:
1482:
1480:
1479:
1474:
1453:
1451:
1450:
1445:
1433:
1431:
1430:
1425:
1401:
1399:
1398:
1393:
1375:
1373:
1372:
1367:
1333:
1331:
1330:
1325:
1320:
1319:
1307:
1306:
1294:
1293:
1281:
1280:
1268:
1267:
1255:
1254:
1211:
1209:
1208:
1203:
1191:
1189:
1188:
1183:
1168:
1166:
1165:
1160:
1158:
1157:
1130:
1128:
1127:
1122:
1110:
1108:
1107:
1102:
1081:
1079:
1078:
1073:
1061:
1059:
1058:
1053:
1038:
1036:
1035:
1030:
1009:
1007:
1006:
1001:
999:
998:
970:
968:
967:
962:
950:
948:
947:
942:
940:
939:
908:
906:
905:
900:
898:
894:
889:
882:
881:
869:
868:
856:
855:
845:
836:
835:
807:
805:
804:
799:
787:
785:
784:
779:
749:
747:
746:
741:
711:
709:
708:
703:
664:
662:
661:
656:
621:
620:
591:
589:
588:
583:
581:
580:
498:
496:
495:
490:
382:axes are called
277:angular momentum
273:angular velocity
234:Earth's rotation
228:can be called a
195:chaotic rotation
168:axis of rotation
143:
136:
132:
129:
123:
121:
80:
56:
48:
21:
4315:
4314:
4310:
4309:
4308:
4306:
4305:
4304:
4270:
4269:
4268:
4256:
4246:
4244:
4234:
4232:
4222:
4220:
4208:
4198:
4196:
4184:
4172:
4164:
4103:
4100:
4095:
4094:
4084:
4082:
4069:
4068:
4064:
4057:
4038:
4037:
4033:
4026:
4007:
4006:
3997:
3961:
3960:
3956:
3930:
3929:
3925:
3918:
3905:
3904:
3900:
3892:Brannon, R.M.,
3891:
3887:
3878:
3876:
3874:
3859:
3858:
3851:
3846:
3841:
3776:Circular motion
3765:
3595:
3588:
3559:amusement rides
3555:
3553:Amusement rides
3510:flight dynamics
3498:
3492:
3490:Flight dynamics
3453:
3447:
3411:
3405:
3362:oblate spheroid
3358:Earth's gravity
3309:
3297:
3291:Rotation period
3268:
3220:
3214:
3212:Euler rotations
3192:. According to
3155:laws of physics
3151:
3120:plane of motion
3112:Euler's theorem
3081:circular motion
3057:
3051:Circular motion
3047:
3045:Circular motion
3035:right-hand rule
2979:
2951:
2945:
2911:
2910:
2882:
2865:
2859:
2812:
2785:
2770:
2725:
2710:
2693:
2686:
2685:
2652:
2651:
2630:
2613:
2606:
2605:
2563:
2546:
2539:
2538:
2510:
2509:
2471:
2454:
2439:
2412:
2397:
2359:
2344:
2331:
2310:
2303:
2302:
2281:
2280:
2252:
2251:
2247:
2199:
2198:
2179:
2178:
2135:
2134:
2100:
2099:
2068:
2067:
2048:
2047:
2014:
2009:
2008:
1975:
1970:
1969:
1945:
1940:
1939:
1897:
1892:
1891:
1890:because, since
1846:
1821:
1787:
1771:
1716:
1703:
1692:
1691:
1645:
1644:
1610:
1609:
1581:
1580:
1556:
1550:
1526:
1525:
1506:
1505:
1485:
1484:
1456:
1455:
1436:
1435:
1407:
1406:
1378:
1377:
1340:
1339:
1311:
1298:
1285:
1272:
1259:
1246:
1214:
1213:
1194:
1193:
1174:
1173:
1149:
1138:
1137:
1113:
1112:
1084:
1083:
1064:
1063:
1044:
1043:
1012:
1011:
990:
973:
972:
953:
952:
931:
914:
913:
873:
860:
847:
846:
840:
824:
813:
812:
790:
789:
770:
769:
766:
760:
717:
716:
673:
672:
612:
607:
606:
575:
574:
563:
551:
550:
536:
520:
508:
507:
481:
480:
475:for the matrix
465:
459:
446:
441:
427:rotation matrix
291:
285:
232:; for example,
144:
133:
127:
124:
81:
79:
69:
57:
46:
35:
28:
23:
22:
15:
12:
11:
5:
4313:
4311:
4303:
4302:
4297:
4292:
4287:
4282:
4272:
4271:
4267:
4266:
4254:
4242:
4230:
4218:
4206:
4194:
4182:
4162:
4161:
4155:
4141:
4135:
4129:
4119:
4099:
4098:External links
4096:
4093:
4092:
4062:
4055:
4031:
4024:
3995:
3974:(3): 417–422.
3954:
3923:
3916:
3898:
3885:
3872:
3848:
3847:
3845:
3842:
3840:
3839:
3838:– spinning toy
3833:
3828:
3823:
3818:
3812:
3806:
3801:
3796:
3790:
3784:
3778:
3773:
3766:
3764:
3761:
3718:two-and-a-half
3714:one-and-a-half
3670:baton twirling
3662:figure skating
3587:
3584:
3572:Chair-O-Planes
3554:
3551:
3494:Main article:
3491:
3488:
3449:Main article:
3446:
3443:
3407:Main article:
3404:
3401:
3308:
3305:
3277:caused by the
3267:
3264:
3216:Main article:
3213:
3210:
3150:
3147:
3046:
3043:
2997:per time), or
2978:
2975:
2944:
2941:
2924:
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2292:{\textstyle t}
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1554:Rotation plane
1552:Main article:
1549:
1548:Rotation plane
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762:Main article:
759:
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529:
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523:
518:
515:
488:
458:
455:
445:
442:
334:Mathematically
323:plane of orbit
287:Main article:
284:
281:
216:is known as a
214:center of mass
206:
146:
145:
60:
58:
51:
43:Rotate, Kansas
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4312:
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4298:
4296:
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4265:
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4219:
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4207:
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4149:
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4127:
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4112:
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4097:
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3691:
3687:
3683:
3679:
3675:
3671:
3667:
3663:
3659:
3654:
3652:
3649:sports, etc.
3648:
3644:
3640:
3636:
3632:
3628:
3624:
3620:
3616:
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3604:
3600:
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3552:
3550:
3548:
3544:
3540:
3535:
3533:
3529:
3528:
3524:
3520:
3516:are known as
3515:
3511:
3502:
3497:
3489:
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3478:
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3378:
3374:
3369:
3367:
3363:
3359:
3355:
3351:
3347:
3342:
3340:
3339:tidal locking
3335:
3333:
3329:
3325:
3324:Doppler shift
3321:
3317:
3313:
3306:
3304:
3302:
3296:
3292:
3284:
3280:
3276:
3272:
3265:
3263:
3261:
3257:
3253:
3248:
3246:
3245:line of nodes
3242:
3238:
3237:line of nodes
3233:
3224:
3219:
3211:
3209:
3207:
3203:
3199:
3195:
3191:
3187:
3182:
3179:
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3164:
3160:
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3128:
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3117:
3113:
3109:
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3070:
3061:
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3044:
3042:
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3036:
3032:
3028:
3023:
3021:
3020:
3015:
3010:
3008:
3004:
3000:
2996:
2992:
2988:
2984:
2976:
2974:
2972:
2969:, where each
2968:
2964:
2960:
2956:
2950:
2942:
2940:
2919:
2907:
2892:
2886:
2883:
2872:
2866:
2860:
2845:
2834:
2825:
2816:
2813:
2797:
2786:
2780:
2774:
2771:
2757:
2740:
2726:
2720:
2714:
2711:
2700:
2694:
2683:
2680:
2660:
2634:
2631:
2620:
2614:
2604:showing that
2591:
2588:
2579:
2573:
2567:
2564:
2553:
2547:
2518:
2506:
2487:
2481:
2475:
2472:
2461:
2455:
2449:
2443:
2440:
2424:
2413:
2401:
2398:
2384:
2378:
2369:
2360:
2354:
2348:
2345:
2338:
2322:
2311:
2300:
2286:
2260:
2244:
2242:
2240:
2236:
2231:
2229:
2207:
2184:
2155:
2149:
2146:
2140:
2114:
2108:
2105:
2096:
2076:
2053:
2027:
2015:
1994:
1979:
1955:
1946:
1919:
1901:
1874:
1871:
1859:
1847:
1843:
1840:
1825:
1818:
1809:
1791:
1784:
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1765:
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1697:
1690:
1689:
1688:
1665:
1659:
1656:
1650:
1624:
1618:
1615:
1589:
1578:
1573:
1570:
1565:
1560:
1555:
1547:
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1531:
1511:
1490:
1470:
1467:
1464:
1461:
1441:
1421:
1418:
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1389:
1386:
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1360:
1354:
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1269:
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1199:
1179:
1170:
1154:
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1118:
1098:
1095:
1092:
1089:
1069:
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1026:
1023:
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1017:
995:
991:
987:
984:
981:
978:
958:
936:
932:
928:
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922:
919:
895:
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883:
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865:
861:
857:
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848:
841:
837:
832:
829:
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821:
818:
811:
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809:
795:
775:
765:
757:
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753:
737:
734:
731:
728:
725:
722:
699:
696:
693:
690:
687:
684:
681:
678:
671:
670:
669:
652:
649:
646:
643:
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634:
631:
628:
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617:
613:
605:
604:
603:
602:
598:
577:
571:
568:
565:
560:
557:
554:
547:
544:
541:
538:
533:
530:
527:
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516:
513:
506:
505:
504:
502:
486:
478:
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469:
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456:
454:
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440:
436:
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428:
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408:
404:
399:
397:
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389:
385:
381:
377:
373:
368:
366:
362:
361:
356:
351:
348:
345:
343:
339:
335:
328:
324:
319:
311:
304:
302:
295:
290:
282:
280:
278:
274:
269:
267:
266:
265:orbital poles
261:
257:
253:
252:Earth's orbit
249:
248:
243:
239:
235:
231:
227:
223:
219:
215:
210:
208:
204:
200:
196:
192:
188:
184:
183:
178:
174:
170:
169:
164:
160:
152:
142:
139:
131:
120:
117:
113:
110:
106:
103:
99:
96:
92:
89: –
88:
84:
83:Find sources:
77:
73:
67:
66:
61:This article
59:
55:
50:
49:
44:
40:
39:Rotate (song)
33:
19:
4252:Solar System
4126:cut-the-knot
4108:
4083:. Retrieved
4074:
4065:
4045:
4034:
4014:
3971:
3967:
3957:
3943:(9): 80–88.
3940:
3936:
3926:
3907:
3901:
3888:
3877:. Retrieved
3862:
3740:
3738:
3729:
3721:
3717:
3713:
3701:
3686:snowboarding
3681:
3677:
3673:
3665:
3657:
3655:
3651:Table tennis
3639:spin bowling
3622:
3618:
3614:
3598:
3596:
3563:Ferris wheel
3556:
3536:
3526:
3522:
3518:
3507:
3481:dwarf planet
3463:, including
3461:Solar System
3454:
3430:
3426:
3422:
3418:
3414:
3412:
3370:
3364:; a similar
3343:
3336:
3310:
3298:
3259:
3249:
3240:
3232:Euler angles
3229:
3218:Euler angles
3183:
3167:
3152:
3134:
3131:rigid bodies
3124:
3097:
3089:rigid bodies
3085:displacement
3066:
3024:
3019:axial vector
3017:
3011:
2980:
2971:intersection
2952:
2908:
2684:
2681:
2507:
2301:
2248:
2238:
2234:
2232:
2227:
2097:
1889:
1576:
1574:
1568:
1563:
1561:
1557:
1404:
1335:
1171:
1041:
911:
808:is found by
767:
751:
714:
667:
594:
476:
470:
466:
451:
447:
411:Euler angles
395:
391:
387:
383:
379:
375:
371:
369:
359:
354:
352:
349:
346:
332:
298:
270:
263:
259:
245:
241:
236:defines the
229:
225:
222:autorotation
221:
217:
211:
199:orientations
193:, including
187:solid figure
180:
173:plane figure
167:
166:
162:
158:
157:
134:
125:
115:
108:
101:
94:
82:
70:Please help
65:verification
62:
4240:Outer space
4228:Spaceflight
4180:Mathematics
3710:waterskiing
3647:flying disc
3631:curve balls
3547:gimbal lock
3530:. The term
3350:gravitation
3332:outer gases
3281:during the
3275:Star trails
3174:homogeneous
3139:translation
3104:orientation
3100:orientation
3093:orientation
3077:orientation
3016:vector (an
2989:(rad/s) or
2963:hypervolume
2299:for which:
2098:This means
1483:, in which
595:A standard
473:eigenvector
342:translation
329:(for Earth)
283:Mathematics
254:around the
4300:Kinematics
4274:Categories
4105:"Rotation"
3981:1607.01102
3879:2023-07-27
3844:References
3749:basketball
3706:gymnastics
3704:, etc. in
3698:somersault
3684:, etc. in
3423:revolution
3415:revolution
3403:Revolution
3373:precession
3252:precession
3186:Lagrangian
3049:See also:
2947:See also:
668:which has
597:eigenvalue
435:quaternion
431:axis angle
415:rigid body
401:See also:
338:rigid body
327:axial tilt
242:revolution
128:March 2014
98:newspapers
87:"Rotation"
4204:Astronomy
4115:EMS Press
4085:8 October
3741:spin move
3543:joysticks
3397:Pole Star
3381:gyroscope
3379:. Like a
3301:astronomy
3266:Astronomy
3196:, if the
3178:isotropic
2991:frequency
2923:→
2876:^
2849:→
2829:^
2801:→
2761:^
2744:→
2704:→
2664:→
2624:^
2583:^
2574:⋅
2557:^
2522:→
2491:^
2482:⋅
2465:→
2428:→
2408:⇒
2388:→
2379:⋅
2373:→
2326:→
2264:→
2211:¯
2159:¯
2150:−
2118:¯
2080:¯
2031:¯
1983:¯
1923:¯
1905:¯
1863:¯
1844:−
1829:¯
1813:¯
1795:¯
1785:−
1754:¯
1745:−
1724:¯
1669:¯
1660:−
1628:¯
1593:¯
1468:−
1355:α
1349:
1309:−
1283:−
1257:−
1232:α
1226:
1200:α
1155:∘
1144:α
1119:α
1024:−
996:∘
988:≤
985:α
982:≤
937:∘
929:≤
926:α
923:≤
884:−
838:
830:−
819:α
776:α
735:±
732:≠
729:θ
726:
700:θ
697:
688:±
685:θ
682:
638:θ
635:
629:λ
623:−
614:λ
572:θ
569:
561:θ
558:
548:θ
545:
539:−
534:θ
531:
487:θ
297:Rotation
226:spin axis
177:clockwise
4280:Rotation
4079:Archived
3937:Computer
3763:See also
3753:football
3666:twirling
3635:baseball
3607:backspin
3568:carousel
3532:rotation
3439:galaxies
3431:rotation
3419:rotation
3377:nutation
3328:sunspots
3256:nutation
3135:rotation
439:isometry
250:), e.g.
159:Rotation
18:Rotating
4264:Science
4192:Physics
4166:Portals
4117:, 2001
3809:Rolling
3781:Cyclone
3759:, etc.
3643:cricket
3615:English
3603:topspin
3539:gimbals
3459:in the
3457:planets
3354:equator
3316:planets
3069:objects
2977:Physics
2967:volumes
360:inverse
355:reverse
112:scholar
4053:
4022:
3914:
3896:, 2018
3870:
3757:tennis
3745:hockey
3730:möbius
3726:diving
3722:gainer
3619:follow
3611:tennis
3586:Sports
3541:, and
3477:Uranus
3413:While
3258:, and
3198:action
3003:torque
2999:period
2682:From:
1968:, and
501:matrix
437:, and
114:
107:
100:
93:
85:
4216:Stars
3976:arXiv
3672:, or
3578:. In
3557:Many
3519:pitch
3484:Pluto
3473:Venus
3465:Earth
3455:Most
3385:Earth
3312:Stars
3285:time
3200:(the
3039:screw
2995:turns
365:group
247:orbit
205:fixed
119:JSTOR
105:books
4087:2013
4051:ISBN
4020:ISBN
3912:ISBN
3868:ISBN
3702:heli
3694:roll
3690:flip
3658:spin
3623:draw
3621:and
3605:and
3599:spin
3525:and
3523:roll
3475:and
3395:and
3375:and
3307:Spin
3293:and
3176:and
3153:The
3137:and
3053:and
3012:The
2981:The
2197:and
2133:and
2066:and
2007:and
1643:and
378:and
325:and
244:(or
230:pole
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