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A **rotation** is a circular movement of an object around a *center* (or *point*) *of rotation*. A three-dimensional object rotates always around an imaginary line called a *rotation axis*. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or *spin*. A **rotation** about an external point, e.g. the Earth about the Sun, is called a revolution or *orbital revolution*, typically when it is produced by gravity.
Mathematically, a rotation is a rigid body movement which, unlike a translation, 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 the axis of the rotation lies external of the body in question then the body is said to orbit. There is no fundamental difference between a “rotation” and a “orbit” and or "spin". The key distinction is simply where the axis of the rotation lies, either within or without a body in question. This distinction can be demonstrated for both “rigid” and “non rigid” bodies.
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. 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, the principal rotations are known as *roll*, and *pitch*, *yaw* (known as Tait-Bryan angles). This terminology is also used in computer graphics.
In astronomy, rotation is a commonly observed phenomenon. Stars, planets and similar bodies all spin around on their 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 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 23.44 degrees, but this angle changes slowly (over thousands of years). (See also Precession of the equinoxes and Pole star.)
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.
Most planets 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.
The speed of rotation is given by the angular frequency (rad/s) or frequency (turns/s, turns/min), or 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.
In flight dynamics, the principal rotations are known as *yaw* and *roll*, *pitch*. 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.
Many amusement rides 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 inversions the rotation about the horizontal axis is one or more full cycles, where inertia keeps people in their seats.
Rotation, usually called *spin*, plays a role in many sports. *Topspin* and *backspin* in tennis. *English*, *follow* and *draw* in billiards and pool. Curve balls in baseball and spin bowling in cricket. Table tennis paddles are specialized to allow players to spin the ball as they hit.

In computing, a **window** is a visual area containing some kind of user interface. It usually has a rectangular shape that can overlap with the area of other windows. It displays the **output** of and may allow **input** to one or more processes.
Windows are primarily associated with graphical displays, where they can be manipulated with a pointer.
A graphical user interface (GUI) using windows as one of its main "metaphors" is called a windowing system.
santoshkamble@pai-ims==Properties==
Windows are two dimensional objects arranged on a plane called the desktop. In a modern full-featured windowing system they can be resized, moved, hidden, restored or closed.
Windows usually include other graphical objects, possibly including a menu-bar, toolbars, controls, icons and often a working area. In the working area, the document, image, folder contents or other main object is displayed. Around the working area, within the bounding window, there may be other smaller window areas, sometimes called panes or panels, showing relevant information or options. The working area of a single document interface holds only one main object. "Child windows" in multiple document interfaces, and tabs for example in many web browsers, can make several similar documents or main objects available within a single main application window.
Some windows in Mac OS X have a feature called a drawer, which is a pane that slides out the side of the window and to show extra options. A window can also be defined as an area within a frame on a computer screen in which a particular program is operated or displayed.
The idea was developed at the Stanford Research Institute (led by Douglas Engelbart). Their earliest systems supported multiple windows, but there was no obvious way to indicate boundaries between them (such as window borders, title bars, etc.).
Research continued at Xerox Corporation's Palo Alto Research Center / PARC (led by Alan Kay). They used overlapping windows.
During the 1980s the term "WIMP", which stands for window, icon, menu, pointer, was coined at PARC.
Apple had worked with PARC briefly at that time. Apple developed an interface based on PARC's interface. It was first used on Apple's Lisa and later Macintosh computers. Microsoft was developing office applications for the "Mac" at that time. Some speculate that this gave them access to Apple's OS before it was released and thus influenced the design of the windowing system in what would eventually be called Windows.
The part of a windowing system which manages window operations is called a *window manager*.
Examples of some current windowing systems:

Generally speaking, an object with **rotational symmetry**, also known in biological contexts as **radial symmetry**, is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted. The degree of rotational symmetry is how many degrees the shape has to be turned to look the same on a different side or vertex. It can not be the same side or vertex.
Formally, rotational symmetry is symmetry with respect to some or all rotations in *m*-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation. Therefore a symmetry group of rotational symmetry is a subgroup of *E*+(*m*) (see Euclidean group).
Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, so space is homogeneous, and the symmetry group is the whole *E*(*m*). With the modified notion of symmetry for vector fields the symmetry group can also be *E*+(*m*).
For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the special orthogonal group SO(*m*), the group of *m*×*m* orthogonal matrices with determinant 1. For this is the rotation group SO(3).
In another meaning of the word, the rotation group *of an object* is the symmetry group within *E*+(*n*), the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.
Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. Because of Noether's theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. See also Rotational invariance.
**Rotational symmetry of order ***n*, also called *n*-fold rotational symmetry, or **discrete rotational symmetry of the ***n*th order, with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of 360°/n (180°, 120°, 90°, 72°, 60°, 51 °, etc.) does not change the object. Note that "1-fold" symmetry is no symmetry, and "2-fold" is the simplest symmetry, so it does not mean "more than basic".
The notation for *n*-fold symmetry is **C**_{n} or simply "*n*". The actual symmetry group is specified by the point or axis of symmetry, together with the *n*. For each point or axis of symmetry the abstract group type is cyclic group Z_{n} of order *n*. Although for the latter also the notation *C*_{n} is used, the geometric and abstract *C*_{n} should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see cyclic symmetry groups in 3D.
The fundamental domain is a sector of 360°/n.
Examples without additional reflection symmetry:
*C*_{n} is the rotation group of a regular *n*-sided polygon in 2D and of a regular *n*-sided pyramid in 3D.
If there is e.g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°.
A typical 3D object with rotational symmetry (possibly also with perpendicular axes) but no mirror symmetry is a propeller.
Double Pendulum fractal
The starting position in shogi
Roundabout traffic sign
Snoldelev Stone's interlocked drinking horns design
Decorative Hindu form of the swastika
For discrete symmetry with multiple symmetry axes through the same point, there are the following possibilities:
In the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges. The other axes are through opposite vertices and through centers of opposite faces, except in the case of the tetrahedron, where the 3-fold axes are each through one vertex and the center of one face.
Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry. The fundamental domain is a half-line.
In three dimensions we can distinguish **cylindrical symmetry** and **spherical symmetry** (no change when rotating about one axis, or for any rotation). That is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, respectively. **Axisymmetric** or **axisymmetrical** are adjectives which refer to an object having cylindrical symmetry, or **axisymmetry**. An example of approximate spherical symmetry is the Earth (with respect to density and other physical and chemical properties).
In 4D, continuous or discrete rotational symmetry about a plane corresponds to corresponding 2D rotational symmetry in every perpendicular plane, about the point of intersection. An object can also have rotational symmetry about two perpendicular planes, e.g. if it is the Cartesian product of two rotationally symmetry 2D figures, as in the case of e.g. the duocylinder and various regular duoprisms.
2-fold rotational symmetry together with single translational symmetry is one of the Frieze groups. There are two rotocenters per primitive cell.
Together with double translational symmetry the rotation groups are the following wallpaper groups, with axes per primitive cell:
Scaling of a lattice divides the number of points per unit area by the square of the scale factor. Therefore the number of 2-, 3-, 4-, and 6-fold rotocenters per primitive cell is 4, 3, 2, and 1, respectively, again including 4-fold as a special case of 2-fold, etc.
3-fold rotational symmetry at one point and 2-fold at another one (or ditto in 3D with respect to parallel axes) implies rotation group p6, i.e. double translational symmetry and 6-fold rotational symmetry at some point (or, in 3D, parallel axis). The translation distance for the symmetry generated by one such pair of rotocenters is 2√3 times their distance.

A **context menu** (also called **contextual**, **shortcut**, and **popup** or **pop-up** **menu**) is a menu in a graphical user interface (GUI) that appears upon user interaction, such as a right-click mouse operation. A context menu offers a limited set of choices that are available in the current state, or context, of the operating system or application. Usually the available choices are actions related to the selected object.
Context menus first appeared in the Smalltalk environment on the Xerox Alto computer, where they were called *pop-up menus.* The NEXTSTEP operating system further developed the idea, incorporating a feature whereby the right or middle mouse button brought the main menu (which was vertical and automatically changed depending on context) to the location of the mouse, thereby eliminating the need to move the mouse pointer all the way across the large (for the time) NextStep screen.
Context menus are opened via various forms of user interaction that target a region of the GUI that supports context menus. The specific form of user interaction and the means by which a region is targeted vary:
Windows 7 changed the mouse click behavior such that the context menu doesn't open while the mouse button is pressed, but only opens the menu when the button is released, so the user has to click again (this time with the first mouse button) to select a context menu item. This behavior differs from Windows XP, Mac OS X, and most Linux distributions.
Context menus are sometimes hierarchically organized, allowing navigation through different levels of the menu structure. The implementations differ: Microsoft Word was one of the first applications to only show sub-entries of some menu entries after clicking an arrow icon on the context menu, otherwise executing an action associated with the parent entry. This makes it possible to quickly repeat an action with the parameters of the previous execution, and to better separate options from actions.
The following window managers provide context menu functionality:
Context menus have received some criticism from usability analysts when improperly used, as some applications make certain features *only* available in context menus, which may confuse even experienced users (especially when the context menus can only be activated in a limited area of the application's client window).
Context menus usually open in a fixed position under the pointer, but when the pointer is near a screen edge the menu will be displaced - thus reducing consistency and impeding use of muscle memory. If the context menu is being triggered by keyboard, such as by using Shift + F10, the context menu appears near the focused widget instead of the position of the pointer, to save recognition efforts.
Microsoft's guidelines call for always using the term *context menu*, and explicitly deprecate *shortcut menu*.

The **Euler angles** are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required. They can be given in several ways, Euler angles being one of them; see charts on SO(3) for others. Euler angles are also used to represent the orientation of a frame of reference (typically, a coordinate system or basis) relative to another. They are typically denoted as α, β, γ, or .
Euler angles also represent a sequence of three *elemental* rotations, i.e. rotations about the axes of a coordinate system. For instance, a first rotation about *z* by an angle α, a second rotation about *x* by an angle β, and a last rotation again about *z*, by an angle γ. These rotations start from a known standard orientation. In physics, this standard initial orientation is typically represented by a motionless (*fixed*, *global*, or *world*) coordinate system. In linear algebra, by a standard basis.
Any orientation can be achieved by composing three elemental rotations. The elemental rotations can either occur about the axes of the fixed coordinate system (extrinsic rotations) or about the axes of a rotating coordinate system, which is initially aligned with the fixed one, and modifies its orientation after each elemental rotation (intrinsic rotations). The rotating coordinate system may be imagined to be rigidly attached to a rigid body. In this case, it is sometimes called a *local* coordinate system. Without considering the possibility of using two different conventions for the definition of the rotation axes (intrinsic or extrinsic), there exist twelve possible sequences of rotation axes, divided in two groups:
Tait-Bryan angles are also called **Cardan angles**, **nautical angles**, **heading, elevation, and bank**, or **yaw, pitch, and roll**. Sometimes, both kinds of sequences are called "Euler angles". In that case, the sequences of the first group are called *proper* or *classic* Euler angles.
Euler angles are a means of representing the spatial orientation of any reference frame (coordinate system or basis) as a composition of three elemental rotations starting from a known standard orientation, represented by another frame (sometimes referred to as the *original* or *fixed* reference frame, or standard basis). The reference orientation can be imagined to be an initial orientation from which the frame virtually rotates to reach its actual orientation. In the following, the axes of the original frame are denoted as *x*,*y*,*z* and the axes of the rotated frame are denoted as *X*,*Y*,*Z*. In geometry and physics, the rotated coordinate system is often imagined to be rigidly attached to a rigid body. In this case, it is called a "local" coordinate system, and it is meant to represent both the position and the orientation of the body.
The geometrical definition (referred sometimes as static) of the Euler angles is based on the axes of the above-mentioned (original and rotated) reference frames and an additional axis called the line of nodes. The line of nodes (*N*) is defined as the intersection of the *xy* and the *XY* coordinate planes. In other words, it is a line passing through the origin of both frames, and perpendicular to the *zZ* plane, on which both *z* and *Z* lie. The three Euler angles are defined as follows:
This definition implies that:
If β is zero, there is no rotation about *N*. As a consequence, *Z* coincides with *z*, α and γ represent rotations about the same axis (*z*), and the final orientation can be obtained with a single rotation about *z*, by an angle equal to α+γ.
The rotated frame *XYZ* may be imagined to be initially aligned with *xyz*, before undergoing to the three elemental rotations represented by Euler angles. Its successive orientations may be denoted as follows:
For the above-listed sequence of rotations, the line of nodes *N* can be simply defined as the orientation of *X* after the first elemental rotation. Hence, *N* can be simply denoted *x’*. Moreover, since the third elemental rotation occurs about *Z*, it does not change the orientation of *Z*. Hence *Z* coincides with *z″*. This allows us to simplify the definition of the Euler angles as follows:
Different authors may use different sets of rotation axes to define Euler angles, or different names for the same angles. Therefore any discussion employing Euler angles should always be preceded by their definition. Unless otherwise stated, this article will use the convention described above.
The three elemental rotations may occur either about the axes *xyz* of the original coordinate system, which is assumed to remain motionless (extrinsic rotations), or about the axes of the rotating coordinate system *XYZ*, which changes its orientation after each elemental rotation (intrinsic rotations). The definition above uses intrinsic rotations.
There are six possibilities of choosing the rotation axes for proper Euler angles. In all of them, the first and third rotation axis are the same. The six possible sequences are:
Euler angles between two reference frames are defined only if both frames have the same handedness.
Angles are commonly defined according to the right hand rule. Namely, they have positive values when they represent a rotation that appears counter-clockwise when observed from a point laying on the positive part of the rotation axis, and negative values when the rotation appears clockwise. The opposite convention (left hand rule) is less frequently adopted.
About the ranges:
The angles α, β and γ are uniquely determined except for the singular case that the *xy* and the *XY* planes are identical, the *z* axis and the *Z* axis having the same or opposite directions. Indeed, if the *z*-axis and the Z-axis are the same, β = 0 and only (α + γ) is uniquely defined (not the individual values), and, similarly, if the *z*-axis and the *Z*-axis are opposite, β = π and only (α − γ) is uniquely defined (not the individual values). These ambiguities are known as gimbal lock in applications.
The fastest way to get the Euler Angles of a given frame is to write the three given vectors as columns of a matrix and compare it with the expression of the theoretical matrix (see later table of matrices). Hence the three Euler Angles can be calculated. Nevertheless, the same result can be reached avoiding matrix algebra, which is more geometrical. Assuming a frame with unitary vectors (*X*, *Y*, *Z*) as in the main diagram, it can be seen that
And, since
we have
As is the double projection of a unitary vector,
There is a similar construction for , projecting it first over the plane defined by the axis *z* and the line of nodes. As the angle between the planes is and , this leads to:
and finally, using the inverse cosine function,
It is interesting to note that the inverse cosine function yields two possible values for the argument. In this geometrical description only one of the solutions is valid. When Euler Angles are defined as a sequence of rotations, all the solutions can be valid, but there will be only one inside the angle ranges. This is because the sequence of rotations to reach the target frame is not unique if the ranges are not previously defined.
For computational purposes, it may be useful to represent the angles using atan2(y,x):
The second type of formalism is called **Tait–Bryan angles**, after Peter Guthrie Tait and George H. Bryan.
The definitions and notations used for Tait-Bryan angles are similar to those described above for proper Euler angles (Classic definition, Alternative definition). The only difference is that Tait–Bryan angles represent rotations about three distinct axes (e.g. *x-y-z*, or *x-y’-z″*), while proper Euler angles use the same axis for both the first and third elemental rotations (e.g., *z-x-z*, or *z-x’-z″*).
This implies a different definition for the line of nodes. In the first case it was defined as the intersection between two homologous Cartesian planes (parallel when Euler angles are zero; e.g. *xy* and *XY*). In the second one, it is defined as the intersection of two non-homologous planes (perpendicular when Euler angles are zero; e.g. *xy* and *YZ*).
The three elemental rotations may occur either about the axes of the original coordinate system, which remains motionless (extrinsic rotations), or about the axes of the rotating coordinate system, which changes its orientation after each elemental rotation (intrinsic rotations).
There are six possibilities of choosing the rotation axes for Tait–Bryan angles. The six possible sequences are:
Tait-Bryan angles are also known as **nautical angles**, because they can be used to describe the orientation of a ship or aircraft, or **Cardan angles**, after the Italian mathematician and physician Gerolamo Cardano (French: ; 24 September 1501 – 21 September 1576) who first described in detail the Cardan suspension and the Cardan joint.
They are also called **heading, elevation and bank**, or **yaw, pitch and roll**. Notice that the second set of terms is also used for the three aircraft principal axes.
Intrinsic rotations are elemental rotations that occur about the axes of the rotating coordinate system *XYZ*, which changes its orientation after each elemental rotation. The *XYZ* system rotates, while *xyz* is fixed. Starting with *XYZ* overlapping *xyz*, a composition of three intrinsic rotations can be used to reach any target orientation for *XYZ*. The Euler or Tait Bryan angles (α, β, γ) are the amplitudes of these elemental rotations. For instance, the target orientation can be reached as follows:
The above-mentioned notation allows us to summarize this as follows: the three elemental rotations of the XYZ-system occur about *z*, *x*’ and *z*″. Indeed, this sequence is often denoted *z-x’-z″*. Sets of rotation axes associated with both proper Euler angles and Tait Bryan angles are commonly named using this notation (see above for details). Sometimes, the same sequence is simply called *z-x-z*, *Z-X-Z*, or *3-1-3*, but this notation may be ambiguous as it may be identical to that used for extrinsic rotations. In this case, it becomes necessary to separately specify whether the rotations are intrinsic or extrinsic.
Rotation matrices can be used to represent a sequence of intrinsic rotations. For instance,
represents a composition of intrinsic rotations about axes *x-y’-z″*, if used to pre-multiply column vectors, while
represents exactly the same composition when used to post-multiply row vectors. See Ambiguities in the definition of rotation matrices for more details.
Extrinsic rotations are elemental rotations that occur about the axes of the fixed coordinate system *xyz*. The *XYZ* system rotates, while *xyz* is fixed. Starting with *XYZ* overlapping *xyz*, a composition of three extrinsic rotations can be used to reach any target orientation for *XYZ*. The Euler or Tait Bryan angles (α, β, γ) are the amplitudes of these elemental rotations. For instance, the target orientation can be reached as follows:
In sum, the three elemental rotations occur about *z*, *x* and *z*. Indeed, this sequence is often denoted *z-x-z* (or 3-1-3). Sets of rotation axes associated with both proper Euler angles and Tait Bryan angles are commonly named using this notation (see above for details).
Rotation matrices can be used to represent a sequence of extrinsic rotations. For instance,
represents a composition of extrinsic rotations about axes *x-y-z*, if used to pre-multiply column vectors, while
represents exactly the same composition when used to post-multiply row vectors. See Ambiguities in the definition of rotation matrices for more details.
Any extrinsic rotation is equivalent to an intrinsic rotation by the same angles but with inverted order of elemental rotations, and vice-versa. For instance, the intrinsic rotations *x-y’-z″* by angles α, β, γ are equivalent to the extrinsic rotations *z-y-x* by angles γ, β, α. Both are represented by a matrix
if *R* is used to pre-multiply column vectors, and by a matrix
if *R* is used to post-multiply row vectors. See Ambiguities in the definition of rotation matrices for more details.
Euler rotations are 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 around the external axis z, the second rotates around the line of nodes and the third one is an intrinsic rotation around an axis fixed in the body that moves.
These rotations are called precession, nutation, and intrinsic rotation (spin). As an example, consider a top. The top spins around its own axis of symmetry; this corresponds to its intrinsic rotation. It also rotates around its pivotal axis, with its center of mass orbiting the pivotal axis; this rotation is a precession. Finally, the top can wobble up and down; the inclination angle is the nutation angle. While all three are rotations when applied over individual frames, only precession is valid as a rotation operator, and only precession can be expressed in general as a matrix in the basis of the space.
If we suppose a set of frames, able to move each with respect to the former according to just one angle, like a gimbal, there will exist an external fixed frame, one final frame and two frames in the middle, which are called "intermediate frames". The two in the middle work as two gimbal rings that allow the last frame to reach any orientation in space.
The gimbal rings indicate some intermediate frames. They can be defined statically too. Taking some vectors **i**, **j** and **k** over the axes x, y and z, and vectors **I**, **J**, **K** over X, Y and Z, and a vector **N** over the line of nodes, some intermediate frames can be defined using the vector cross product, as following:
These intermediate frames are equivalent to those of the gimbal. They are such that they differ from the previous one in just a single elemental rotation. This proves that:
Euler angles are one way to represent orientations. There are others, and it is possible to change to and from other conventions.
Any orientation can be achieved by composing three elemental rotations, starting from a known standard orientation. Equivalently, any rotation matrix *R* can be decomposed as a product of three elemental rotation matrices. For instance:
is a rotation matrix that may be used to represent a composition of intrinsic rotations about axes *x-y’-z″*. However, both the definition of the elemental rotation matrices *X*, *Y*, *Z*, and their multiplication order depend on the choices taken by the user about the definition of both rotation matrices and Euler angles (see, for instance, Ambiguities in the definition of rotation matrices). Unfortunately, different sets of conventions are adopted by users in different contexts. The following table was built according to this set of conventions:
For the sake of simplicity, the following table uses the following nomenclature:
To change the formulas for the opposite direction of rotation, change the signs of the sine functions. To change the formulas for passive rotations, transpose the matrices (then each matrix transforms the initial coordinates of a vector remaining fixed to the coordinates of the same vector measured in the rotated reference system; same rotation axis, same angles, but now the coordinate system rotates, rather than the vector).
Unit quaternions, also known as Euler–Rodrigues parameters, provide another mechanism for representing 3D rotations. This is equivalent to the special unitary group description.
Expressing rotations in 3D as unit quaternions instead of matrices has some advantages:
Other representation comes from the Geometric algebra(GA). GA is a higher level abstraction, in which the quaternions are an even subalgebra. The principal tool in GA is the rotor where angle of rotation, rotation axis (unitary vector) and pseudoscalar (trivector in )
The Euler angles form a chart on all of SO(3), the special orthogonal group of rotations in 3D space. The chart is smooth except for a polar coordinate style singularity along β=0. See charts on SO(3) for a more complete treatment.
The space of rotations is called in general "The Hypersphere of rotations", though this is a misnomer: the group Spin(3) is isometric to the hypersphere *S*3, but the rotation space SO(3) is instead isometric to the real projective space **RP**3 which is a 2-fold quotient space of the hypersphere. This 2-to-1 ambiguity is the mathematical origin of spin in physics.
A similar three angle decomposition applies to SU(2), the special unitary group of rotations in complex 2D space, with the difference that β ranges from 0 to 2π. These are also called Euler angles.
The Haar measure for Euler angles has the simple form sin(β).dα.dβ.dγ, usually normalized by a factor of 1/8π².
For example, to generate uniformly randomized orientations, let α and γ be uniform from 0 to 2π, let *z* be uniform from −1 to 1, and let β = arccos(*z*).
It is possible to define parameters analogous to the Euler angles in dimensions higher than three.
The number of degrees of freedom of a rotation matrix is always less than the dimension of the matrix squared. That is, the elements of a rotation matrix are not all completely independent. For example, the rotation matrix in dimension 2 has only one degree of freedom, since all four of its elements depend on a single angle of rotation. A rotation matrix in dimension 3 (which has nine elements) has three degrees of freedom, corresponding to each independent rotation, for example by its three Euler angles or a magnitude one (unit) quaternion.
In SO(4) the rotation matrix is defined by two quaternions, and is therefore 6-parametric (three degrees of freedom for every quaternion). The 4x4 rotation matrices have therefore 6 out of 16 independent components.
Any set of 6 parameters that define the rotation matrix could be considered an extension of Euler angles to dimension 4.
In general, the number of euler angles in dimension D is quadratic in D; since any one rotation consists of choosing two dimensions to rotate between, the total number of rotations available in dimension is , which for yields .
Their main advantage over other orientation descriptions is that they are directly measurable from a gimbal mounted in a vehicle. As gyroscopes keep their rotation axis constant, angles measured in a gyro frame are equivalent to angles measured in the lab frame. Therefore gyros are used to know the actual orientation of moving spacecraft, and Euler angles are directly measurable. Intrinsic rotation angle cannot be read from a single gimbal, so there has to be more than one gimbal in a spacecraft. Normally there are at least three for redundancy. There is also a relation to the well-known gimbal lock problem of Mechanical Engineering .
The most popular application is to describe aircraft attitudes, normally using a Tait–Bryan convention so that zero degrees elevation represents the horizontal attitude. Tait–Bryan angles represent the orientation of the aircraft respect a reference axis system (*world frame*) with three angles which in the context of an aircraft are normally called Heading, Elevation and Bank. When dealing with vehicles, different axes conventions are possible.
When studying rigid bodies in general, one calls the *xyz* system *space coordinates*, and the *XYZ* system *body coordinates*. The space coordinates are treated as unmoving, while the body coordinates are considered embedded in the moving body. Calculations involving acceleration, angular acceleration, angular velocity, angular momentum, and kinetic energy are often easiest in body coordinates, because then the moment of inertia tensor does not change in time. If one also diagonalizes the rigid body's moment of inertia tensor (with nine components, six of which are independent), then one has a set of coordinates (called the principal axes) in which the moment of inertia tensor has only three components.
The angular velocity of a rigid body takes a simple form using Euler angles in the moving frame. Also the Euler's rigid body equations are simpler because the inertia tensor is constant in that frame.
Euler angles, normally in the Tait–Bryan convention, are also used in robotics for speaking about the degrees of freedom of a wrist. They are also used in Electronic stability control in a similar way.
Gun fire control systems require corrections to gun-order angles (bearing and elevation) to compensate for deck tilt (pitch and roll). In traditional systems, a stabilizing gyroscope with a vertical spin axis corrects for deck tilt, and stabilizes the optical sights and radar antenna. However, gun barrels point in a direction different from the line of sight to the target, to anticipate target movement and fall of the projectile due to gravity, among other factors. Gun mounts roll and pitch with the deck plane, but also require stabilization. Gun orders include angles computed from the vertical gyro data, and those computations involve Euler angles.
Euler angles are also used extensively in the quantum mechanics of angular momentum. In quantum mechanics, explicit descriptions of the representations of *SO(3)* are very important for calculations, and almost all the work has been done using Euler angles. In the early history of quantum mechanics, when physicists and chemists had a sharply negative reaction towards abstract group theoretic methods (called the *Gruppenpest*), reliance on Euler angles was also essential for basic theoretical work.
In materials science, crystallographic texture (or preferred orientation) can be described using Euler angles. In texture analysis, the Euler angles provide the necessary mathematical depiction of the orientation of individual crystallites within a polycrystalline material, allowing for the quantitative description of the macroscopic material. The most common definition of the angles is due to Bunge and corresponds to the ZXZ convention. It is important to note, however, that the application generally involves axis transformations of tensor quantities, i.e. passive rotations. Thus the matrix that corresponds to the Bunge Euler angles is the transpose of that shown in the table above.
Many mobile computing devices contain accelerometers which can determine these devices' Euler angles with respect to the earth's gravitational attraction. These are used in applications such as games, bubble level simulations, and kaleidoscopes.][

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 point. A rotation is different from a translation, which has no fixed points, and from a reflection, which "flips" the bodies it is transforming. A rotation and the above-mentioned transformations are isometries; they leave the distance between any two points unchanged after the transformation.
It is important to know the frame of reference when considering rotations, as all rotations are described relative to a particular frame of reference. In general for any orthogonal transformation on a body in a coordinate system there is an inverse transformation which if applied to the frame of reference results in the body being at the same coordinates. For example in two dimensions rotating a body clockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed.
Only a single angle is needed to specify a rotation in two dimensions – the *angle of rotation*. To calculate the rotation two methods can be used, either matrix algebra or complex numbers. In each the rotation is acting to rotate an object counterclockwise through an angle *θ* about the origin.
To carry out a rotation using matrices the point (*x*, *y*) to be rotated is written as a vector, then multiplied by a matrix calculated from the angle, , like so:
where (*x*′, *y*′) are the co-ordinates of the point after rotation, and the formulae for *x*′ and *y*′ can be seen to be
The vectors and have the same magnitude and are separated by an angle as expected.
Points can also be rotated using complex numbers, as the set of all such numbers, the complex plane, is geometrically a two dimensional plane. The point (*x*, *y*) in the plane is represented by the complex number
This can be rotated through an angle *θ* by multiplying it by e*iθ*, then expanding the product using Euler's formula as follows:
which gives the same result as before, 1
Like complex numbers rotations in two dimensions are commutative, unlike in higher dimensions. They have only one degree of freedom, as such rotations are entirely determined by the angle of rotation.
Rotations in ordinary three-dimensional space differ from those in two dimensions in a number of important ways. Rotations in three dimensions are generally not commutative, so the order in which rotations are applied is important. They have three degrees of freedom, the same as the number of dimensions.
A three dimensional rotation can be specified in a number of ways. The most usual methods are as follows.
As in two dimensions a matrix can be used to rotate a point (*x*, *y*, *z*) to a point (*x*′, *y*′, *z*′). The matrix used is a 3 × 3 matrix,
This is multiplied by a vector representing the point to give the result
The matrix **A** is a member of the three dimensional special orthogonal group, SO(3), that is it is an orthogonal matrix with determinant 1. That it is an orthogonal matrix means that its rows are a set of orthogonal unit vectors (so they are an orthonormal basis) as are its columns, making it simple to spot and check if a matrix is a valid rotation matrix. The determinant of a rotation orthogonal matrix must be 1. The only other possibility for the determinant of an orthogonal matrix is -1, and this result means the transformation is a reflection, improper rotation or inversion in a point, i.e. not a rotation.
Matrices are often used for doing transformations, especially when a large number of points are being transformed, as they are a direct representation of the linear operator. Rotations represented in other ways are often converted to matrices before being used. They can be extended to represent rotations and transformations at the same time using Homogeneous coordinates. Transformations in this space are represented by 4 × 4 matrices, which are not rotation matrices but which have a 3 × 3 rotation matrix in the upper left corner.
The main disadvantage of matrices is that they are more expensive to calculate and do calculations with. Also in calculations where numerical instability is a concern matrices can be more prone to it, so calculations to restore orthonormality, which are expensive to do for matrices, need to be done more often.
One way of generalising the two dimensional angle of rotation is to specify three rotation angles, carried out in turn about the three principal axes. They individually can be labelled yaw, pitch, and roll, but in mathematics are more often known by their mathematical name, Euler angles. They have the advantage of modelling a number of physical systems such as gimbals, 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 where the angles cannot be uniquely calculated for certain rotations.
Euler rotations are a set 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 around the external axis z, the second rotates around the line of nodes and the third one is an intrinsic rotation around an axis fixed in the body that moves.
These rotations are called Precession, Nutation, and intrinsic rotation.
A second way of generalising the two dimensional angle of rotation is to specify an angle with the axis about which the rotation takes place. It can be used to model motion constrained by a hinges and Axles, and so is easily visualised, perhaps even more so than Euler angles. There are two ways to represent it;
Usually the angle and axis pair is easier to work with, while the rotation vector is more compact, requiring only three numbers like Euler angles. But like Euler angles it is usually converted to another representation before being used.
Quaternions are in some ways the least intuitive representation of three dimensional rotations. They are not the three dimensional instance of a general approach, like matrices; nor are they easily related to real world models, like Euler angles or axis angles. But they are more compact than matrices and easier to work with than all other methods, so are often preferred in real world applications.][
A rotation quaternion consists of four real numbers, constrained so the length of the quaternion considered as a vector is 1. This constraint limits the degree of freedom of the quaternion to three, as required. It can be thought of as a generalisation of the complex numbers, by e.g. the Cayley–Dickson construction, and generates rotations in a similar way by multiplication. But unlike matrices and complex numbers two multiplications are needed:
where **q** is the rotation quaternion, **q**−1 is its inverse, and **x** is the vector treated as a quaternion. The quaternion can be related to the rotation vector form of the axis angle rotation by the exponential map over the quaternions,
Where **v** is the rotation vector treated as a quaternion.
A general rotation in four dimensions has only one fixed point, the centre of rotation, and no axis of rotation. Instead the rotation has two mutually orthogonal planes of rotation, each of which is fixed in the sense that points in each plane stay within the planes. The rotation has two angles of rotation, one for each plane of rotation, through which points in the planes rotate. If these are ω_{1} and ω_{2} then all points not in the planes rotate through an angle between ω_{1} and ω_{2}.
If ω_{1} = ω_{2} the rotation is a double rotation and all points rotate through the same angle so any two orthogonal planes can be taken as the planes of rotation. If one of ω_{1} and ω_{2} is zero, one plane is fixed and the rotation is simple. If both ω_{1} and ω_{2} are zero the rotation is the identity rotation.
Rotations in four dimensions can be represented by 4th order orthogonal matrices, as a generalisation of the rotation matrix. Quaternions can also be generalised into four dimensions, as even Multivectors of the four dimensional Geometric algebra. A third approach, which only works in four dimensions, is to use a pair of unit quaternions.
Rotations in four dimensions have six degrees of freedom, most easily seen when two unit quaternions are used, as each has three degrees of freedom (they lie on the surface of a 3-sphere) and 2 × 3 = 6.
One application of this is special relativity, as it can be considered to operate in a four dimensional space, spacetime, spanned by three space dimensions and one of time. In special relativity this space is linear and the four dimensional rotations, called Lorentz transformations, have practical physical interpretations.
If a rotation is only in the three space dimensions, i.e. in a plane that is entirely in space, then this rotation is the same as a spatial rotation in three dimensions. But a rotation in a plane spanned by a space dimension and a time dimension is a hyperbolic rotation, a transformation between two different reference frames, which is sometimes called a "Lorentz boost". These transformations, which are not actual rotations, but squeeze mappings, are sometimes described with Minkowski diagrams. The study of relativity is concerned with the Lorentz group generated by the space rotations and hyperbolic rotations.
The set of all matrices *M(***v**,θ) described above together with the operation of matrix multiplication is the rotation group SO(3).
More generally, coordinate rotations in any dimension are represented by orthogonal matrices. The set of all orthogonal matrices of the *n*-th dimension which describe proper rotations (determinant = +1), together with the operation of matrix multiplication, forms the *special orthogonal group*: SO(*n*).
Orthogonal matrices have real elements. The analogous complex-valued matrices are the unitary matrices. The set of all unitary matrices in a given dimension *n* forms a unitary group of degree *n*, U(n); and the subgroup of U(n) representing proper rotations forms a special unitary group of degree *n*, SU(n). The elements of SU(2) are used in quantum mechanics to rotate spin.

Unit quaternions, also known as versors, provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions. Compared to Euler angles they are simpler to compose and avoid the problem of gimbal lock. Compared to rotation matrices they are more numerically stable and may be more efficient. Quaternions have found their way into applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, and orbital mechanics of satellites.
When used to represent rotation, unit quaternions are also called **rotation quaternions**. When used to represent an orientation (rotation relative to a reference position), they are called **orientation quaternions** or **attitude quaternions**.
According to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or Coordinate system about a fixed point is equivalent to a single rotation by a given angle about a fixed axis (called *Euler axis*) that runs through the fixed point. The Euler axis is typically represented by a unit vector . Therefore, any rotation in three dimensions can be represented as a combination of a vector and a scalar . Quaternions give a simple way to encode this axis–angle representation in four numbers, and to apply the corresponding rotation to a position vector representing a point relative to the origin in 3**R**.
A Euclidean vector such as or can be rewritten as or , where , , are unit vectors representing the three Cartesian axes. A rotation with an angle of rotation of around the axis defined by a unit vector
is represented by a quaternion using an extension of Euler's formula:
The rotation is clockwise if our line of sight points in the same direction as . The halves enable the encoding of both clockwise and counter-clockwise rotations.
It can be shown that this rotation can be applied to an ordinary vector in 3-dimensional space, considered as a quaternion with a real coordinate equal to zero, by evaluating the **Hamilton product**:
where is the new position vector of the point after the rotation, and is the quaternion conjugate of :
This operation is known as conjugation by .
It follows that conjugation by the product of two quaternions is the composition of conjugations by these quaternions. If and are unit quaternions, then rotation (conjugation) by is
which is the same as rotating (conjugating) by and then by . The scalar component of the result is necessarily zero.
The quaternion inverse of a rotation is the opposite rotation, since . The square of a quaternion rotation is a rotation by twice the angle around the same axis. More generally is a rotation by times the angle around the same axis as . This can be extended to arbitrary real , allowing for smooth interpolation between spatial orientations; see Slerp.
Two rotation quaternions can be combined into one equivalent quaternion by the relation:
in which corresponds to the rotation followed by the rotation . (Note that quaternion multiplication is not commutative.) Thus, an arbitrary number of rotations can be composed together and then applied as a single rotation.
Consider the rotation around the axis , with a rotation angle of 120°, or radians.
The length of is , the half angle is (60°) with cosine , () and sine , (). We are therefore dealing with a conjugation by the unit quaternion
If is the rotation function,
It can be proved that the inverse of a unit quaternion is obtained simply by changing the sign of its imaginary components. As a consequence,
and
This can be simplified, using the ordinary rules for quaternion arithmetic, to
As expected, the rotation corresponds to keeping a cube held fixed at one point, and rotating it 120° about the long diagonal through the fixed point (observe how the three axes are permuted cyclically).
Let's show how we reached the previous result. Let's develop the expression of (in two stages), and apply the rules
It gives us:
which is the expected result. As we can see, such computations are relatively long and tedious if done manually; however, in a computer program, this amounts to calling the quaternion multiplication routine twice.
A quaternion rotation can be algebraically manipulated into a quaternion-derived rotation matrix. By simplifying the quaternion multiplications , they can be rewritten as a rotation matrix given an axis–angle representation:
where and is shorthand for and respectively. Although care should be taken (due to degeneracy as the quaternion approaches the identity quaternion or the sine of the angle approaches zero) the axis and angle can be extracted via:
Note that the equality holds only when the square root of the sum of the squared imaginary terms takes the same sign as .
As with other schemes to apply rotations, the centre of rotation must be translated to the origin before the rotation is applied and translated back to its original position afterwards.
The complex numbers can be defined by introducing an abstract symbol which satisfies the usual rules of algebra and additionally the rule . This is sufficient to reproduce all of the rules of complex number arithmetic: for example:
In the same way the quaternions can be defined by introducing abstract symbols , , which satisfy the rules and the usual algebraic rules *except* the commutative law of multiplication (a familiar example of such a noncommutative multiplication is matrix multiplication). From this all of the rules of quaternion arithmetic follow: for example, one can show that:
.
The imaginary part of a quaternion behaves like a vector in three dimension vector space, and the real part behaves like a scalar in . When quaternions are used in geometry, it is more convenient to define them as a scalar plus a vector:
Those who have studied vectors at school might find it strange to add a *number* to a *vector*, as they are objects of very different natures, or to *multiply* two vectors together, as this operation is usually undefined. However, if one remembers that it is a mere notation for the real and imaginary parts of a quaternion, it becomes more legitimate. In other words, the correct reasoning is the addition of two quaternions, one with zero vector/imaginary part, and another one with zero scalar/real part:
We can express quaternion multiplication in the modern language of vector cross and dot products (which were actually inspired by the quaternions in the first place][). In place of the rules we have the quaternion multiplication rule:
where:
Quaternion multiplication is noncommutative (because of the cross product, which anti-commutes), while scalar-scalar and scalar-vector multiplications commute. From these rules it follows immediately that (see details):
The (left and right) multiplicative inverse or reciprocal of a nonzero quaternion is given by the conjugate-to-norm ratio (see details):
as can be verified by direct calculation.
Let be a unit vector (the rotation axis) and let . Our goal is to show that
yields the vector rotated by an angle around the axis . Expanding out, we have
where and are the components of *v*} perpendicular and parallel to *u* respectively. This is the formula of a rotation by around the *u* axis.
A very formal explanation of the properties used in this section is given by Altman.
Unit quaternions represent the group of Euclidean rotations in three dimensions in a very straightforward way. The correspondence between rotations and quaternions can be understood by first visualizing the space of rotations itself.
In order to visualize the space of rotations, it helps to consider a simpler case. Any rotation in three dimensions can be described by a rotation by some angle about some axis; for our purposes, we will use an axis *vector* to establish handedness for our angle. Consider the special case in which the axis of rotation lies in the *xy* plane. We can then specify the axis of one of these rotations by a point on a circle through which the vector crosses, and we can select the radius of the circle to denote the angle of rotation. Similarly, a rotation whose axis of rotation lies in the *xy* plane can be described as a point on a sphere of *fixed* radius in *three* dimensions. Beginning at the north pole of a sphere in three dimensional space, we specify the point at the north pole to be the identity rotation (a zero angle rotation). Just as in the case of the identity rotation, no axis of rotation is defined, and the angle of rotation (zero) is irrelevant. A rotation having a very small rotation angle can be specified by a slice through the sphere parallel to the *xy* plane and very near the north pole. The circle defined by this slice will be very small, corresponding to the small angle of the rotation. As the rotation angles become larger, the slice moves in the negative *z* direction, and the circles become larger until the equator of the sphere is reached, which will correspond to a rotation angle of 180 degrees. Continuing southward, the radii of the circles now become smaller (corresponding to the absolute value of the angle of the rotation considered as a negative number). Finally, as the south pole is reached, the circles shrink once more to the identity rotation, which is also specified as the point at the south pole.
Notice that a number of characteristics of such rotations and their representations can be seen by this visualization. The space of rotations is continuous, each rotation has a neighborhood of rotations which are nearly the same, and this neighborhood becomes flat as the neighborhood shrinks. Also, each rotation is actually represented by two antipodal points on the sphere, which are at opposite ends of a line through the center of the sphere. This reflects the fact that each rotation can be represented as a rotation about some axis, or, equivalently, as a negative rotation about an axis pointing in the opposite direction (a so-called double cover). The "latitude" of a circle representing a particular rotation angle will be half of the angle represented by that rotation, since as the point is moved from the north to south pole, the latitude ranges from zero to 180 degrees, while the angle of rotation ranges from 0 to 360 degrees. (the "longitude" of a point then represents a particular axis of rotation.) Note however that this set of rotations is not closed under composition. Two successive rotations with axes in the *xy* plane will not necessarily give a rotation whose axis lies in the *xy* plane, and thus cannot be represented as a point on the sphere. This will not be the case with a general rotation in 3-space, in which rotations do form a closed set under composition.
This visualization can be extended to a general rotation in 3-dimensional space. The identity rotation is a point, and a small angle of rotation about some axis can be represented as a point on a sphere with a small radius. As the angle of rotation grows, the sphere grows, until the angle of rotation reaches 180 degrees, at which point the sphere begins to shrink, becoming a point as the angle approaches 360 degrees (or zero degrees from the negative direction). This set of expanding and contracting spheres represents a hypersphere in four dimensional space (a 3-sphere). Just as in the simpler example above, each rotation represented as a point on the hypersphere is matched by its antipodal point on that hypersphere. The "latitude" on the hypersphere will be half of the corresponding angle of rotation, and the neighborhood of any point will become "flatter" (i.e. be represented by a 3-D Euclidean space of points) as the neighborhood shrinks. This behavior is matched by the set of unit quaternions: A general quaternion represents a point in a four dimensional space, but constraining it to have unit magnitude yields a three dimensional space equivalent to the surface of a hypersphere. The magnitude of the unit quaternion will be unity, corresponding to a hypersphere of unit radius. The vector part of a unit quaternion represents the radius of the 2-sphere corresponding to the axis of rotation, and its magnitude is the cosine of half the angle of rotation. Each rotation is represented by two unit quaternions of opposite sign, and, as in the space of rotations in three dimensions, the quaternion product of two unit quaternions will yield a unit quaternion. Also, the space of unit quaternions is "flat" in any infinitesimal neighborhood of a given unit quaternion.
We can parameterize the surface of a sphere with two coordinates, such as latitude and longitude. But latitude and longitude are ill-behaved (degenerate) at the north and south poles, though the poles are not intrinsically different from any other points on the sphere. At the poles (latitudes +90° and −90°), the longitude becomes meaningless.
It can be shown that no two-parameter coordinate system can avoid such degeneracy. We can avoid such problems by embedding the sphere in three-dimensional space and parameterizing it with three Cartesian coordinates , placing the north pole at , the south pole at , and the equator at , . Points on the sphere satisfy the constraint , so we still have just two degrees of freedom though there are three coordinates. A point on the sphere represents a rotation in the ordinary space around the horizontal axis directed by the vector by an angle .
In the same way the hyperspherical space of 3D rotations can be parameterized by three angles (Euler angles), but any such parameterization is degenerate at some points on the hypersphere, leading to the problem of gimbal lock. We can avoid this by using four Euclidean coordinates , with . The point represents a rotation around the axis directed by the vector by an angle
The multiplication of quaternions is non-commutative. Since the multiplication of unit quaternions corresponds to the composition of three dimensional rotations, this property can be made intuitive by showing that three dimensional rotations are not commutative in general.
Set two books next to each other. Rotate one of them 90 degrees clockwise around the *z* axis, then flip it 180 degrees around the *x* axis. Take the other book, flip it 180 around *x* axis first, and 90 clockwise around *z* later. The two books do not end up parallel. This shows that, in general, the composition of two different rotations around two distinct spatial axes will not commute.
Quaternions, and quaternion multiplication in particular, do not confer an orientation ("handedness") to space. This may be contrasted with the vector cross product, which does: in a three-dimensional vector space, the three vectors in the equation will always form a right-handed set (or a left-handed set, depending on how the cross product is defined), thus fixing an orientation in the vector space. Quaternion multiplication is symmetrical with respect to the two possible orientations of space, thus does not in itself allow one orientation in space to be chosen above the other.
The representation of a rotation as a quaternion (4 numbers) is more compact than the representation as an orthogonal matrix (9 numbers). Furthermore, for a given axis and angle, one can easily construct the corresponding quaternion, and conversely, for a given quaternion one can easily read off the axis and the angle. Both of these are much harder with matrices or Euler angles.
In video games and other applications, one is often interested in “smooth rotations”, meaning that the scene should slowly rotate and not in a single step. This can be accomplished by choosing a curve such as the spherical linear interpolation in the quaternions, with one endpoint being the identity transformation 1 (or some other initial rotation) and the other being the intended final rotation. This is more problematic with other representations of rotations.
When composing several rotations on a computer, rounding errors necessarily accumulate. A quaternion that’s slightly off still represents a rotation after being normalised: a matrix that’s slightly off may not be orthogonal anymore and is harder to convert back to a proper orthogonal matrix.
Quaternions also avoid a phenomenon called gimbal lock which can result when, for example in pitch/yaw/roll rotational systems, the pitch is rotated 90° up or down, so that yaw and roll then correspond to the same motion, and a degree of freedom of rotation is lost. In a gimbal-based aerospace inertial navigation system, for instance, this could have disastrous results if the aircraft is in a steep dive or ascent.
The orthogonal matrix corresponding to a rotation by the unit quaternion (with ) when post-multiplying with a column vector is given by
One must be careful when converting a rotation matrix to a quaternion, as several straightforward methods tend to be unstable when the trace (sum of the diagonal elements) of the rotation matrix is zero or very small. For a stable method of converting an orthogonal matrix to a quaternion, see Rotation matrix #Quaternion.
The above section described how to recover a quaternion from a 3 × 3 rotation matrix . Suppose, however, that we have some matrix that is not a pure rotation—due to round-off errors, for example—and we wish to find the quaternion that most accurately represents . In that case we construct a symmetric 4 × 4 matrix
and find the eigenvector corresponding to the largest eigenvalue (that value will be 1 if and only if is a pure rotation). The quaternion so obtained will correspond to the rotation closest to the original matrix ]* *[
This section discusses the performance implications of using quaternions versus other methods (axis/angle or rotation matrices) to perform rotations in 3D.
* Note: angle/axis can be stored as 3 elements by multiplying the unit rotation axis by half of the rotation angle, forming the logarithm of the quaternion, at the cost of additional calculations.
There are three basic approaches to rotating a vector *v*:
A pair of unit quaternions and can represent any rotation in 4D space. Given a four dimensional vector *v*, and pretending that it is a quaternion, we can rotate the vector *v* like this:

It is straightforward to check that for each matrix: T, that is, that each matrix (and hence both matrices together) represents a rotation. Note that since , the two matrices must commute. Therefore, there are two commuting subgroups of the set of four dimensional rotations. Arbitrary four dimensional rotations have 6 degrees of freedom, each matrix represents 3 of those 6 degrees of freedom. Since an infinitesimal four-dimensional rotation can be represented by a pair of quaternions (as follows), all (non-infinitesimal) four-dimensional rotations can also be represented. E. P. Battey-Pratt & T. J. Racey (1980) Geometric Model for Fundamental Particles*International Journal of Theoretical Physics. Vol 19, No. 6*

It is straightforward to check that for each matrix: T, that is, that each matrix (and hence both matrices together) represents a rotation. Note that since , the two matrices must commute. Therefore, there are two commuting subgroups of the set of four dimensional rotations. Arbitrary four dimensional rotations have 6 degrees of freedom, each matrix represents 3 of those 6 degrees of freedom. Since an infinitesimal four-dimensional rotation can be represented by a pair of quaternions (as follows), all (non-infinitesimal) four-dimensional rotations can also be represented. E. P. Battey-Pratt & T. J. Racey (1980) Geometric Model for Fundamental Particles

**Euclidean geometry** is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the *Elements*. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The *Elements* begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the *Elements* states results of what are now called algebra and number theory, explained in geometrical language.

For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only where the gravitational field is weak.

**Kinematics** is the branch of classical mechanics which describes the motion of points, bodies (objects) and systems of bodies (groups of objects) without consideration of the causes of motion. The term is the English version of A.M. Ampère's *cinématique,* which he constructed from the Greek κίνημα, **kinema** (movement, motion), derived from κινεῖν, **kinein** (to move).

The study of *kinematics* is often referred to as the *geometry of motion.* (See analytical dynamics for more detail on usage.)

A **rotation** is a circular movement of an object around a *center* (or *point*) *of rotation*. A three-dimensional object rotates always around an imaginary line called a *rotation axis*. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or *spin*. A **rotation** about an external point, e.g. the Earth about the Sun, is called a revolution or *orbital revolution*, typically when it is produced by gravity.

**Mathematical analysis** is a branch of mathematics that includes the theories of differentiation, integration, measure, limits, infinite series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry. However, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).

Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. In India, the 12th century mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem.

30 known species

A **mouse** (plural: **mice**) is a small mammal belonging to the order of rodents, characteristically having a pointed snout, small rounded ears, and a long naked or almost hairless tail. The best known mouse species is the common house mouse (*Mus musculus*). It is also a popular pet. In some places, certain kinds of field mice are also common. This rodent is eaten by large birds such as hawks and eagles. They are known to invade homes for food and occasionally shelter.

In geometry, a **plane of rotation** is an abstract object used to describe or visualise rotations in space. In three dimensions it is an alternative to the axis of rotation, but unlike the axis of rotation it can be used in other dimensions, such as two, four or more dimensions.

Mathematically such planes can be described in a number of ways. They can be described in terms of planes and angles of rotation. They can be associated with bivectors from geometric algebra. They are related to the eigenvalues and eigenvectors of a rotation matrix. And in particular dimensions they are related to other algebraic and geometric properties, which can then be generalised to other dimensions.

**Geometry** (Ancient Greek: *γεωμετρία*; *geo-* "earth", *-metron* "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of a formal mathematical science emerging in the West as early as Thales (6th Century BC). By the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow. Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. Both geometry and astronomy were considered in the classical world to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.

The introduction of coordinates by René Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures, such as plane curves, could now be represented analytically, i.e., with functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. Furthermore, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry. The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry.

**Physics** (from Greek φυσική (ἐπιστήμη), i.e. "knowledge of nature", from φύσις, *physis*, i.e. "nature") is the natural science that involves the study of matter and its motion through space and time, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.

Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy. Over the last two millennia, physics was a part of natural philosophy along with chemistry, certain branches of mathematics, and biology, but during the Scientific Revolution in the 17th century, the natural sciences emerged as unique research programs in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms of other sciences, while opening new avenues of research in areas such as mathematics and philosophy.

Generally speaking, an object with **rotational symmetry**, also known in biological contexts as **radial symmetry**, is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted. The degree of rotational symmetry is how many degrees the shape has to be turned to look the same on a different side or vertex. It can not be the same side or vertex.

Formally, rotational symmetry is symmetry with respect to some or all rotations in *m*-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation. Therefore a symmetry group of rotational symmetry is a subgroup of *E*+(*m*) (see Euclidean group).

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