A sphere rotating around an axis.

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center (or point) of rotation. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume 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. Speed is the rate of motion, or equivalently the rate of change in position often expressed as Distance d traveled per unit of In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position A circular motion about an external point, e. g. the Earth about the Sun, is called an orbit or more properly an orbital revolution. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 The Sun (Sol is the Star at the center of the Solar System. In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star

Mathematics

Main article: Rotation (mathematics)

Rotation of a planar figure around a point

Mathematically, a rotation is, unlike a translation, a rigid body movement which keeps a point fixed. 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 Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Euclidean geometry, a translation is moving every point a constant distance in a specified direction In Physics, a rigid body is an idealization of a solid body of finite size in which Deformation is neglected 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. In Kinematics, Euler's rotation theorem states that in Three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed

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) of a rotation is also a rotation. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to Thus, the rotations around a point/axis form a group. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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.

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). Flight dynamics is the science of air and space vehicle orientation and control in three dimensions The Tait-Bryan rotations, named after Peter Guthrie Tait and George Bryan. This terminology is also used in computer graphics. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data

See also: curl, cyclic permutation, Euler angles, rigid body, rotation around a fixed axis, rotation group, rotation matrix, axis angle, quaternion, and isometry

Astronomy

Relations between rotation axis, plane of orbit and axial tilt (for Earth). cURL is a Command line tool for transferring files with URL syntax. A cyclic Permutation is built from one or more sets of elements in Cyclic order. The Euler angles were developed by Leonhard Euler to describe the orientation of a Rigid body (a body in which the relative position of all its points is constant In Physics, a rigid body is an idealization of a solid body of finite size in which Deformation is neglected Rotational motion can occur around more than one axis at once and can involve phenomena such as wobbling and Precession. This article is about rotations in three-dimensional Euclidean space In Matrix theory, a rotation matrix is a real Square matrix whose Transpose is its inverse and whose Determinant is +1 The axis angle representation of a rotation also known as the exponential coordinates of a rotation parameterizes a Rotation by two values a Unit vector indicating Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician For the Mechanical engineering and Architecture usage see Isometric projection. The orbital plane of an object orbiting another is the geometrical plane in which the orbit is embedded. In Astronomy, axial tilt is the Inclination angle of a planet's rotational axis in relation to its orbital plane.

In astronomy, rotation is a commonly observed phenomenon. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study Stars, planets and similar bodies all spin around on their axes (the plural of axis). A star is a massive luminous ball of plasma. The nearest star to Earth is the Sun, which is the source of most of the Energy on Earth A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is 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. Stellar rotation is the angular motion of a Star about its axis The Doppler effect (or Doppler shift) named after Christian Doppler, is the change in Frequency and Wavelength of a Wave for

This rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect of gravity the closer one is to the equator. The equator (sometimes referred to colloquially as "the Line") is the intersection of the Earth 's surface with the plane perpendicular to the One effect is that an object weighs slightly less at the equator. Another is that the Earth is slightly deformed into an oblate spheroid. An oblate Spheroid is a rotationally symmetric Ellipsoid having a polar axis shorter than the diameter of the equatorial circle whose plane

Another consequence of the rotation of a planet is the phenomenon of precession. Precession refers to a change in the direction of the axis of a rotating object Like a gyroscope, the overall effect is a slight "wobble" in the movement of the axis of a planet. A gyroscope is a device for measuring or maintaining orientation, based on the principles of Angular momentum. Currently the tilt of the Earth's axis to its orbital plane (obliquity of the ecliptic) is 23. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 In Astronomy, axial tilt is the Inclination angle of a planet's rotational axis in relation to its orbital plane. 45 degrees, but this angle changes slowly (over thousands of years). (See also Precession of the equinoxes and Pole star. In Astronomy, Precession refers to the movement of the rotational axis of a body such as a planet with respect to Inertial space. A pole star is a visible star especially a prominent one that is approximately aligned with the Earth 's Axis of rotation; that is a star whose apparent position )

Rotation and revolution

Main article: 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. In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star 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. A galaxy is a massive gravitationally bound system consisting of Stars an Interstellar medium of gas and dust, and Dark matter The motion of the components of galaxies is complex, but it usually includes a rotation component. A galaxy is a massive gravitationally bound system consisting of Stars an Interstellar medium of gas and dust, and Dark matter

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). A separate article treats the phenomenon of Tidal resonance in Oceanography. Far Side of the Moon, in original French, La face cachée de la lune, is a 2003 film by Robert Lepage.

Retrograde rotation

Main article: Retrograde motion#Retrograde rotation

Most planets in our solar system, including Earth, spin in the same direction as they orbit the Sun. Direct motion is the motion of a Planetary body in a direction similar to that of other bodies within its system and is sometimes called prograde motion. A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is The Solar System consists of the Sun and those celestial objects bound to it by Gravity. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 The Sun (Sol is the Star at the center of the Solar System. The exceptions are Venus and Uranus. The VENUS ( V ictoria E xperimental N etwork U nder the S ea project is a cabled sea floor observatory operated by the University 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. A dwarf planet, as defined by the International Astronomical Union (IAU is a Celestial body Orbiting the Sun that is massive enough to be rounded

Physics

Main article: Angular momentum

The speed of rotation is given by the angular frequency (rad/s) or frequency (turns/s, turns/min), or period (seconds, days, etc. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position Do not confuse with Angular velocity In Physics (specifically Mechanics and Electrical engineering) angular frequency Frequency is a measure of the number of occurrences of a repeating event per unit Time. A turn is a unit of plane angle equal to 360° or 2π Radians As an angular unit it is mainly useful for large Angles such as in connection with Coils and Periodicity is the quality of occurring at regular intervals or periods (in Time or Space) and can occur in different contexts A Clock marks ). The time-rate of change of angular frequency is angular acceleration (rad/s²), This change is caused by torque. A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about The ratio of the two (how heavy is it to start, stop, or otherwise change rotation) is given by the moment of inertia. This article is about the moment of inertia of a rotating object.

The angular velocity vector also describes the direction of the axis of rotation. Do not confuse with Angular frequency The unit for angular velocity is rad/s 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. For the related yet different principle relating to electromagnetic coils see Right hand grip rule. A screw is a shaft with a helical groove or thread formed on its surface and provision at one end to turn the screw

See also: rotational energy, angular velocity, Centrifugal force (fictitious), centripetal force, circular motion, circular orbit, Coriolis effect, spin (physics), rotational spectroscopy, and Rigid body dynamics#Rigid body angular momentum

Aviation

In flight dynamics, the principal rotations are known as pitch, roll and yaw. The rotational energy or angular kinetic energy is the Kinetic energy due to the rotation of an object and is part of its total kinetic energy. Do not confuse with Angular frequency The unit for angular velocity is rad/s The centripetal force is the external force required to make a body follow a curved path In Physics, circular motion is Rotation along a Circle: a circular path or a circular Orbit. For other meanings of the term "orbit" see Orbit (disambiguation In Astrodynamics or Celestial mechanics a circular In physics the Coriolis effect is an apparent deflection of moving objects when they are viewed from a Rotating frame of reference. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin Rotational spectroscopy or microwave spectroscopy studies the absorption and emission Electromagnetic radiation (typically in the Microwave Flight dynamics is the science of air and space vehicle orientation and control in three dimensions 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 rides provide rotation. Rides redirects here For the album by Reef see Rides (album, and for the BBC television series see Rides (TV series. A Ferris wheel has a horizontal central axis, and parallel axes for each gondola, where the rotation is opposite, by gravity or mechanically. A Ferris wheel (also known as an observation wheel or big wheel) is a Nonbuilding structure consisting of an upright wheel with passenger gondolas attached 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. A carousel ( carrousel in French) is an Amusement ride consisting of a rotating platform with seats for passengers 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. The Chair-O-Planes, sometimes called a swing carousel or wave swinger, is a Fairground ride that is a variation on the Carousel in which the The centripetal force is the external force required to make a body follow a curved path In roller coaster inversions the rotation about the horizontal axis is one or more full cycles, where inertia keeps people in their seats. A roller coaster inversion is an element of a Roller coaster track that turns riders upside-down and then rights them

Sports

Rotation, usually called spin, plays a role in many sports. Topspin and backspin in tennis. Tennis is a sport played between two players ( singles) or between two teams of two players each ( doubles) English, follow and draw in billiards and pool. Curve balls in baseball and spin bowling in cricket. The curveball is a breaking pitch in Baseball thrown with a grip and hand movement that imparts down and/or sideways spin to the ball. Spin bowling, sometimes known as slow bowling, is a technique used for bowling in the Sport of Cricket. Table tennis paddles are specialized to allow players to spin the ball as they hit it.

External links

• Product of Rotations at cut-the-knot
• When a Triangle is Equilateral at cut-the-knot
• Rotate Points Using Polar Coordinates
• Rotation in Two Dimensions by Sergio Hannibal Mejia after work by Roger Germundsson and Understanding 3D Rotation by Roger Germundsson, The Wolfram Demonstrations Project. Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in Mathematics. Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in Mathematics.