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Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional object has an infinite number of possible central axes and rotational directions. If the rotation axis passes internally through the body's own center of mass, then the body is said to be ''autorotating'' or '' spinning'', and the surface intersection of the axis can be called a ''
pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...
''. A rotation around a completely external axis, e.g. the planet
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
around the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
, is called ''revolving'' or ''
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
ing'', typically when it is produced by
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
, and the ends of the rotation axis can be called the ''
orbital pole An orbital pole is either point at the ends of an imaginary line segment that runs through the center of an orbit (of a revolving body like a planet, moon or satellite) and is perpendicular to the orbital plane. Projected onto the celestial sphe ...
s''.


Mathematics

Mathematically, a rotation is a
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
movement which, unlike a
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
, keeps a point fixed. This definition applies to rotations within both two and three dimensions (in a plane and in space, respectively.) 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 90 degrees 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) of a rotation is also a rotation. Thus, the rotations around a point/axis form a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
. 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. In
flight dynamics Flight dynamics in aviation and spacecraft, is the study of the performance, stability, and control of vehicles flying through the air or in outer space. It is concerned with how forces acting on the vehicle determine its velocity and attit ...
, the principal rotations are known as ''yaw'', ''pitch'', and ''roll'' (known as
Tait–Bryan angles The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref> The ...
). This terminology is also used in
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
.


Astronomy

In
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
, rotation is a commonly observed phenomenon. Stars,
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s and similar bodies all 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 through Doppler shift or by tracking active surface features. This rotation induces a
centrifugal acceleration In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
in the reference frame of the Earth which slightly counteracts the effect of gravitation the closer one is to the equator.
Earth's gravity The gravity of Earth, denoted by , is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation). It is a vector quanti ...
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 oblate spheroid; a similar
equatorial bulge An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a sphere. On E ...
develops for other planets. Another consequence of the rotation of a planet is the phenomenon of
precession Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In oth ...
. 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 Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
's axis to its orbital plane (
obliquity of the ecliptic In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and o ...
) is 23.44 degrees, but this angle changes slowly (over thousands of years). (See also
Precession of the equinoxes In astronomy, axial precession is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's rotational axis. In the absence of precession, the astronomical body's orbit would show axial parallelism. In partic ...
and
Pole star A pole star or polar star is a star, preferably bright, nearly aligned with the axis of a rotating astronomical body. Currently, Earth's pole stars are Polaris (Alpha Ursae Minoris), a bright magnitude-2 star aligned approximately with its ...
.)


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 around 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 galaxies is complex, but it usually includes a rotation component.


Retrograde rotation

Most
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s in the
Solar System The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
, including
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
, spin in the same direction as they orbit the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
. The exceptions are
Venus Venus is the second planet from the Sun. It is sometimes called Earth's "sister" or "twin" planet as it is almost as large and has a similar composition. As an interior planet to Earth, Venus (like Mercury) appears in Earth's sky never f ...
and
Uranus Uranus is the seventh planet from the Sun. Its name is a reference to the Greek god of the sky, Uranus ( Caelus), who, according to Greek mythology, was the great-grandfather of Ares (Mars), grandfather of Zeus (Jupiter) and father of ...
. 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
dwarf planet A dwarf planet is a small planetary-mass object that is in direct orbit of the Sun, smaller than any of the eight classical planets but still a world in its own right. The prototypical dwarf planet is Pluto. The interest of dwarf planets to ...
Pluto Pluto (minor-planet designation: 134340 Pluto) is a dwarf planet in the Kuiper belt, a ring of bodies beyond the orbit of Neptune. It is the ninth-largest and tenth-most-massive known object to directly orbit the Sun. It is the largest ...
(formerly considered a planet) is anomalous in several ways, including that it also rotates on its side.


Physics

The speed of rotation is given by the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
(rad/s) or
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
( turns per time), or
period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Periodic sentence (or rhetorical period), a concept ...
(seconds, days, etc.). The time-rate of change of angular frequency is angular acceleration (rad/s²), caused by
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
. 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 (an ''
axial vector In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its o ...
'') also describes the direction of the axis of rotation. Similarly the torque is an axial vector. The physics of the
rotation around a fixed axis Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rot ...
is mathematically described with the
axis–angle representation In mathematics, the axis–angle representation of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector indicating the direction of an axis of rotation, and an angle describing the magnitu ...
of rotations. 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 A screw and a bolt (see '' Differentiation between bolt and screw'' below) are similar types of fastener typically made of metal and characterized by a helical ridge, called a ''male thread'' (external thread). Screws and bolts are used to f ...
.


Cosmological principle

The
laws of physics Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) ...
are currently believed to be invariant under any fixed rotation. (Although they do appear to change when viewed from a rotating viewpoint: see rotating frame of reference.) In modern physical cosmology, the
cosmological principle In modern physical cosmology, the cosmological principle is the notion that the spatial distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throu ...
is the notion that the distribution of matter in the universe is homogeneous and isotropic 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. In particular, for a system which behaves the same regardless of how it is oriented in space, its
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
is rotationally invariant. According to
Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
, if the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
(the
integral over time In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...
of its Lagrangian) of a physical system is invariant under rotation, then angular momentum is conserved.


Euler rotations

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 Euler 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 the
line of nodes An orbital node is either of the two points where an orbit intersects a plane of reference to which it is inclined. A non-inclined orbit, which is contained in the reference plane, has no nodes. Planes of reference Common planes of refere ...
around the external axis ''z'', the second rotates around the
line of nodes An orbital node is either of the two points where an orbit intersects a plane of reference to which it is inclined. A non-inclined orbit, which is contained in the reference plane, has no nodes. Planes of reference Common planes of refere ...
and the third one is an intrinsic rotation around an axis fixed in the body that moves. These rotations are called
precession Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In oth ...
,
nutation Nutation () is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behaviour of a mechanism. In an appropriate reference frame ...
, and ''intrinsic rotation''.


Flight dynamics

In
flight dynamics Flight dynamics in aviation and spacecraft, is the study of the performance, stability, and control of vehicles flying through the air or in outer space. It is concerned with how forces acting on the vehicle determine its velocity and attit ...
, the principal rotations described with Euler angles above 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. Principal rotations have the advantage of modelling a number of physical systems such as
gimbal A gimbal is a pivoted support that permits rotation of an object about an axis. A set of three gimbals, one mounted on the other with orthogonal pivot axes, may be used to allow an object mounted on the innermost gimbal to remain independent of ...
s, and joysticks, 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
gimbal lock Gimbal lock is the loss of one degree of freedom in a three-dimensional, three-gimbal mechanism that occurs when the axes of two of the three gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate t ...
where the angles cannot be uniquely calculated for certain rotations.


Amusement rides

Many
amusement ride Amusement rides, sometimes called carnival rides, are mechanical devices or structures that move people especially kids to create fun and enjoyment. Rides are often perceived by many as being scary or more dangerous than they actually are. This ...
s provide rotation. A
Ferris wheel A Ferris wheel (also called a Giant Wheel or an observation wheel) is an amusement ride consisting of a rotating upright wheel with multiple passenger-carrying components (commonly referred to as passenger cars, cabins, tubs, gondolas, capsule ...
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 swing ride or chair swing ride (sometimes called a swing carousel, wave swinger, yo-yo, waver swinger, Chair-O-Planes, Dodo or swinger) is an amusement ride that is a variation on the carousel in which the seats are suspended from the ro ...
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 A roller coaster inversion is a roller coaster element in which the track turns riders upside-down and then returns them to an upright position. Early forms of inversions were circular in nature and date back to 1848 on the Centrifugal railway in ...
s the rotation about the horizontal axis is one or more full cycles, where inertia keeps people in their seats.


Sports

Rotation of a ball or other object, usually called ''spin'', plays a role in many sports, including
topspin In ball sports, topspin or overspin is a property of a ball that rotates forwards as it is moving. Topspin on a ball propelled through the air imparts a downward force that causes the ball to drop, due to its interaction with the air (see Magn ...
and
backspin In racquet sports and golf, backspin or underspin refers to the reverse rotation of a ball, in relation to the ball's trajectory, that is imparted on the ball by a slice or chop shot. Backspin generates an upward force that lifts the ball (see M ...
in
tennis Tennis is a racket sport that is played either individually against a single opponent ( singles) or between two teams of two players each ( doubles). Each player uses a tennis racket that is strung with cord to strike a hollow rubber ball ...
, ''English'', ''follow'' and ''draw'' in billiards and pool,
curve ball In baseball and softball, the curveball is a type of pitch thrown with a characteristic grip and hand movement that imparts forward spin to the ball, causing it to dive as it approaches the plate. Varieties of curveball include the 12–6 cur ...
s in
baseball Baseball is a bat-and-ball sport played between two teams of nine players each, taking turns batting and fielding. The game occurs over the course of several plays, with each play generally beginning when a player on the fielding t ...
,
spin bowling Spin bowling is a bowling technique in cricket, in which the ball is delivered slowly but with the potential to deviate sharply after bouncing. The bowler is referred to as a spinner. Purpose The main aim of spin bowling is to bowl the cricket ...
in
cricket Cricket is a bat-and-ball game played between two teams of eleven players on a field at the centre of which is a pitch with a wicket at each end, each comprising two bails balanced on three stumps. The batting side scores runs by str ...
,
flying disc A frisbee (pronounced ), also called a flying disc or simply a disc, is a gliding toy or sporting item that is generally made of injection-molded plastic and roughly in diameter with a pronounced lip. It is used recreationally and competitiv ...
sports, etc.
Table tennis Table tennis, also known as ping-pong and whiff-whaff, is a sport in which two or four players hit a lightweight ball, also known as the ping-pong ball, back and forth across a table using small solid rackets. It takes place on a hard table div ...
paddles are manufactured with different surface characteristics to allow the player to impart a greater or lesser amount of spin to the ball. Rotation of a player one or more times around a vertical axis may be called ''spin'' in
figure skating Figure skating is a sport in which individuals, pairs, or groups perform on figure skates on ice. It was the first winter sport to be included in the Olympic Games, when contested at the 1908 Olympics in London. The Olympic disciplines are m ...
, ''twirling'' (of the baton or the performer) in
baton twirling Baton twirling involves using the body to spin a metal rod in a coordinated routine. It is similar to rhythmic gymnastics or color guard. Description Twirling combines dance, agility, coordination and flexibility while manipulating a single bat ...
, or ''360'', ''540'', ''720'', etc. in snowboarding, etc. Rotation of a player or performer one or more times around a horizontal axis may be called a flip,
roll Roll or Rolls may refer to: Movement about the longitudinal axis * Roll angle (or roll rotation), one of the 3 angular degrees of freedom of any stiff body (for example a vehicle), describing motion about the longitudinal axis ** Roll (aviation) ...
,
somersault A somersault (also ''flip'', ''heli'', and in gymnastics ''salto'') is an acrobatic exercise in which a person's body rotates 360° around a horizontal axis with the feet passing over the head. A somersault can be performed forwards, backwards ...
, ''heli'', etc. in
gymnastics Gymnastics is a type of sport that includes physical exercises requiring balance, strength, flexibility, agility, coordination, dedication and endurance. The movements involved in gymnastics contribute to the development of the arms, legs, s ...
,
waterskiing Water skiing (also waterskiing or water-skiing) is a surface water sport in which an individual is pulled behind a boat or a cable ski installation over a body of water, skimming the surface on two skis or one ski. The sport requires suffici ...
, or many other sports, or a ''one-and-a-half'', ''two-and-a-half'', ''gainer'' (starting facing away from the water), etc. in
diving Diving most often refers to: * Diving (sport), the sport of jumping into deep water * Underwater diving, human activity underwater for recreational or occupational purposes Diving or Dive may also refer to: Sports * Dive (American football), a ...
, etc. A combination of vertical and horizontal rotation (back flip with 360°) is called a ''möbius'' in waterskiing freestyle jumping. Rotation of a player around a vertical axis, generally between 180 and 360 degrees, may be called a ''spin move'' and is used as a deceptive or avoidance maneuver, 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
hockey Hockey is a term used to denote a family of various types of both summer and winter team sports which originated on either an outdoor field, sheet of ice, or dry floor such as in a gymnasium. While these sports vary in specific rules, numbers o ...
,
basketball Basketball is a team sport in which two teams, most commonly of five players each, opposing one another on a rectangular court, compete with the primary objective of shooting a basketball (approximately in diameter) through the defender's h ...
, football of various codes,
tennis Tennis is a racket sport that is played either individually against a single opponent ( singles) or between two teams of two players each ( doubles). Each player uses a tennis racket that is strung with cord to strike a hollow rubber ball ...
, etc.


Fixed axis vs. fixed point

The ''end result'' of any sequence of rotations of any object in 3D about a fixed point is always equivalent to a rotation about an axis. However, an object may ''physically'' rotate in 3D about a fixed point on more than one axis simultaneously, in which case there is no single fixed axis of rotation - just the fixed point. However, these two descriptions can be reconciled - such a physical motion can always be re-described in terms of a single axis of rotation, provided the orientation of that axis relative to the object is allowed to change moment by moment.


Axis of 2 dimensional rotations

2 dimensional rotations, unlike the 3 dimensional ones, possess no axis of rotation. 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 In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
for the matrix A representing the rotation. Every 2D rotation around the origin through an angle \theta in counterclockwise direction can be quite simply represented by the following matrix: :A = \begin \cos \theta & -\sin \theta\\ \sin \theta & \cos \theta \end A standard
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
determination leads to the characteristic equation :\lambda^2 -2 \lambda \cos \theta + 1 = 0, which has :\cos \theta \pm i \sin \theta as its eigenvalues. Therefore, there is no real eigenvalue whenever \cos \theta \neq \pm 1, meaning that no real vector in the plane is kept unchanged by A.


Rotation angle and axis in 3 dimensions

Knowing that the trace is an invariant, the rotation angle \alpha for a proper orthogonal 3x3 rotation matrix A is found by \alpha=\cos^\left(\frac\right) Using the principal arc-cosine, this formula gives a rotation angle satisfying 0\le\alpha\le 180^\circ. 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 m can always be written as a rotation having 0\le\alpha\le 180^\circ if the axis is replaced with n=-m.) Every proper rotation A in 3D space has an axis of rotation, which is defined such that any vector v that is aligned with the rotation axis will not be affected by rotation. Accordingly, A v = v , 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 \alpha 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 \alpha=180^\circ, 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 n denotes the unit eigenvector aligned with the rotation axis, and if \alpha denotes the rotation angle, then it can be shown that 2\sin(\alpha)n=\. 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 \sin(\alpha)=0. 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 A+I that has a nonzero magnitude.Brannon, R.M.
"Rotation, Reflection, and Frame Change"
2018
This discussion applies to a proper rotation, and hence \det A = 1. Any improper orthogonal 3x3 matrix B may be written as B=-A, in which A 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 A is also the eigenvector of B corresponding to an eigenvalue of -1.


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 don't 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, \bar is also an eigenvector, and v + \bar and i(v - \bar) are such that their scalar product vanishes: : i (v^\text + \bar^\text)(v - \bar) = i (v^\text v - \bar^\text \bar + \bar^\text v - v^\text \bar ) = 0 because, since \bar^\text \bar is real, it equals its complex conjugate v^\text v , and \bar^\text v and v^\text \bar are both representations of the same scalar product between v and \bar . This means v + \bar and i(v - \bar) are orthogonal vectors. Also, they are both real vectors by construction. These vectors span the same subspace as v and \bar , 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''.


See also

* * * * * * * , the fastest rotation object * * * * * * *


References


External links

*
Product of Rotations
at cut-the-knot. cut-the-knot.org
When a Triangle is Equilateral
at cut-the-knot. cut-the-knot.org
Rotate Points Using Polar Coordinates
howtoproperly.com
Rotation in Two Dimensions
by Sergio Hannibal Mejia after work by Roger Germundsson an
Understanding 3D Rotation
by Roger Germundsson,
Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
. demonstrations.wolfram.com
Rotation, Reflection, and Frame Change: Orthogonal tensors in computational engineering mechanics
IOP Publishing {{Authority control Euclidean geometry Classical mechanics Orientation (geometry) Kinematics