Rotation 39789 222197671 2008-06-28T01:13:08Z MFNickster 61685 /* Mathematics */ update wl to 'curl (mathematics)' instead of dab page {{otheruses4|movement of a physical body}} {{Unreferenced|date=April 2008}} [[Image:Rotating Sphere.gif|right|thumb|A sphere rotating around an axis.]] A '''rotation''' is a movement of an object in a circular motion. A two-[[dimension]]al object rotates around a ''center'' (or ''[[point (geometry)|point]]'') ''of rotation''. A three-dimensional object rotates around a line called an ''axis''. If the axis of rotation is within the body, the body is said to rotate upon itself, or ''spin''—which implies relative [[speed]] and perhaps free-movement with [[angular momentum]]. A circular motion about an external point, e.g. the [[Earth]] about the [[Sun]], is called an ''[[orbit]]'' or more properly an ''[[orbital revolution]].'' ==Mathematics== {{Main|Rotation (mathematics)}} [[Image:Rotation illustration.png|right|thumb|Rotation of a planar figure around a point]] [[Mathematics|Mathematically]], a rotation is, unlike a [[translation (geometry)|translation]], a [[rigid body]] movement which keeps a point fixed. This definition applies to rotations within both two and three dimensions (in a plane and in space, respectively.) A rotation in three-dimensional space keeps an entire line fixed, i.e. a rotation in three-dimensional space is a rotation around an axis. This follows from [[Euler's rotation theorem]]. All rigid body movements are rotations, translations, or combinations of the two. 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. [[Image:Flight dynamics with text.png|right|thumb|The principal axes of rotation in space]] 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. In [[flight dynamics]], the principal rotations are known as ''pitch'', ''roll'' and ''yaw'' (known as [[Tait-Bryan angles]]). This terminology is also used in [[computer graphics]]. {{see also|curl (mathematics)|cyclic permutation|Euler angles|rigid body|rotation around a fixed axis|rotation group|rotation matrix|axis angle|quaternion|isometry}} ==Astronomy==<!-- This section is linked from [[Redshift]] --> [[Image:AxialTiltObliquity.png|thumb|right|Relations between rotation axis, [[Orbital plane (astronomy)|plane of orbit]] and [[Axial tilt|axial tilt]] (for Earth).]] In [[astronomy]], rotation is a commonly observed phenomenon. [[Star]]s, [[planet]]s and similar bodies all spin around on their axes (the plural of ''axis''). The rotation rate of planets in the solar system was first measured by tracking visual features. [[Stellar rotation]] is measured through [[Doppler shift]] or by tracking active surface features. This rotation induces a [[Centrifugal force (fictitious)|centrifugal acceleration]] in the reference frame of the Earth which slightly counteracts the effect of gravity the closer one is to the [[equator]]. One effect is that an object weighs slightly less at the equator. Another is that the Earth is slightly deformed into an [[oblate spheroid]]. Another consequence of the rotation of a planet is the phenomenon of [[precession]]. Like a [[gyroscope]], the overall effect is a slight "wobble" in the movement of the axis of a planet. Currently the tilt of the [[Earth]]'s axis to its orbital plane ([[obliquity of the ecliptic]]) is 23.45 degrees, but this angle changes slowly (over thousands of years). (See also [[Precession of the equinoxes]] and [[Pole star]].) ===Rotation and revolution=== {{Main|Orbital revolution}} While revolution is often used as a synonym for rotation, in many fields, particularly astronomy and related fields, revolution, often referred to as orbital revolution for clarity, is used when one body moves around another while rotation is used to mean the movement around an axis. Moons revolve about their planet, planets revolve about their star (such as the Earth around the Sun); and stars slowly revolve about their [[galaxial center]]. The motion of the components of [[galaxy|galaxies]] is complex, but it usually includes a rotation component. The Moon makes one complete rotation during one complete orbital revolution around the Earth (an effect called [[tidal locking]]) so that the same side of the Moon always faces the Earth (the other side is called the [[far side of the Moon]]). ===Retrograde rotation=== {{Main|Retrograde motion#Retrograde rotation}} Most [[planet]]s in our [[solar system]], including [[Earth]], spin in the same direction as they orbit the [[Sun]]. The exceptions are [[Venus]] and [[Uranus]]. 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. Venus may be thought of as rotating slowly backwards (or being "upside down"). The [[dwarf planet]] [[Pluto]] (formerly considered a planet) is anomalous in this and other ways. ==Physics== {{Main|Angular momentum}} The speed of rotation is given by the [[angular frequency]] (rad/s) or [[frequency]] ([[turn (geometry)|turns]]/s, turns/min), or [[periodicity|period]] (seconds, days, etc.). The time-rate of change of angular frequency is angular acceleration (rad/s²), This change is caused by [[torque]]. The ratio of the two (how heavy is it to start, stop, or otherwise change rotation) is given by the [[moment of inertia]]. The [[angular velocity]] ''vector'' also describes the direction of the axis of rotation. Similarly the torque is a vector. According to the [[right-hand rule]], the direction away from the observer is associated with clockwise rotation and the direction towards the observer with counterclockwise rotation, like a [[screw]]. {{See also|rotational energy|angular velocity|Centrifugal force (fictitious)|centripetal force|circular motion|circular orbit|Coriolis effect|spin (physics)|rotational spectroscopy|Rigid body dynamics#Rigid body angular momentum}} ==Aviation== In [[flight dynamics]], the principal rotations are known as ''pitch'', ''roll'' and ''yaw''. The term rotation is also used in aviation to refer to the upward pitch (nose moves up) of an aircraft, particularly when starting the climb after takeoff. ==Amusement rides== Many [[amusement ride]]s provide rotation. A [[Ferris 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. A [[carousel]] provides rotation about a vertical axis. Many rides provide a combination of rotations about several axes. In [[Chair-O-Planes]] the rotation about the vertical axis is provided mechanically, while the rotation about the horizontal axis is due to the [[centripetal force]]. In [[Roller coaster inversion|roller coaster inversions]] the rotation about the horizontal axis is one or more full cycles, where inertia keeps people in their seats. ==Sports== Rotation, usually called ''spin'', plays a role in many sports. ''Topspin'' and ''backspin'' in [[tennis]]. ''English'', ''follow'' and ''draw'' in [[Billiards#Shooting techniques/mechanics|billiards and pool]]. [[Curve ball]]s in baseball and [[spin bowling]] in cricket. [[Table tennis]] paddles are specialized to allow players to spin the ball as they hit it. ==External links== * [http://www.cut-the-knot.org/Curriculum/Geometry/RotationTransform.shtml Product of Rotations] at [[cut-the-knot]] * [http://www.cut-the-knot.org/Curriculum/Geometry/Connes.shtml When a Triangle is Equilateral] at [[cut-the-knot]] * [http://howtoproperly.com/rotate-points-using-polar-coordinates Rotate Points Using Polar Coordinates] * [http://demonstrations.wolfram.com/RotationInTwoDimensions/ Rotation in Two Dimensions] by Sergio Hannibal Mejia after work by Roger Germundsson and [http://demonstrations.wolfram.com/Understanding3DRotation/ Understanding 3D Rotation] by Roger Germundsson, [[The Wolfram Demonstrations Project]]. 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