Angular displacement of a body is the

__all__ rotations may be represented in this form. The product $\backslash mathbf\backslash theta$ is the "generator" of the particular rotation, being the vector (''x'',''y'',''z'') associated with the matrix A. This shows that the rotation matrix and the axis-angle format are related by the exponential function.
One can derive a simple expression for the generator G. One starts with an arbitrary plane defined by a pair of perpendicular unit vectors a and b. In this plane one can choose an arbitrary vector x with perpendicular y. One then solves for y in terms of x and substituting into an expression for a rotation in a plane yields the rotation matrix R which includes the generator G = ba^{T} − ab^{T}.
:$\backslash begin\; x\; \&=\; a\; \backslash cos\backslash left(\; \backslash alpha\; \backslash right)\; +\; b\; \backslash sin\backslash left(\; \backslash alpha\; \backslash right)\; \backslash \backslash \; y\; \&=\; -a\; \backslash sin\backslash left(\; \backslash alpha\; \backslash right)\; +\; b\; \backslash cos\backslash left(\; \backslash alpha\; \backslash right)\; \backslash \backslash \; \backslash cos\backslash left(\; \backslash alpha\; \backslash right)\; \&=\; a^T\; x\; \backslash \backslash \; \backslash sin\backslash left(\; \backslash alpha\; \backslash right)\; \&=\; b^T\; x\; \backslash \backslash \; y\; \&=\; -ab^T\; x\; +\; ba^T\; x\; =\; \backslash left(\; ba^T\; -\; ab^T\; \backslash right)x\; \backslash \backslash \; \backslash \backslash \; x\text{\'}\; \&=\; x\; \backslash cos\backslash left(\; \backslash beta\; \backslash right)\; +\; y\; \backslash sin\backslash left(\; \backslash beta\; \backslash right)\; \backslash \backslash \; \&=\; \backslash left;\; href="/html/ALL/s/I\_\backslash cos\backslash left(\_\backslash beta\_\backslash right)\_+\_\backslash left(\_ba^T\_-\_ab^T\_\backslash right)\_\backslash sin\backslash left(\_\backslash beta\_\backslash right)\_\backslash right.html"\; ;"title="I\; \backslash cos\backslash left(\; \backslash beta\; \backslash right)\; +\; \backslash left(\; ba^T\; -\; ab^T\; \backslash right)\; \backslash sin\backslash left(\; \backslash beta\; \backslash right)\; \backslash right">I\; \backslash cos\backslash left(\; \backslash beta\; \backslash right)\; +\; \backslash left(\; ba^T\; -\; ab^T\; \backslash right)\; \backslash sin\backslash left(\; \backslash beta\; \backslash right)\; \backslash right$
To include vectors outside the plane in the rotation one needs to modify the above expression for R by including two projection operators that partition the space. This modified rotation matrix can be rewritten as an

Conversely, a

{{Classical mechanics derived SI units
Angle

angle
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

in radian
The radian, denoted by the symbol \text, is the SI unit
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

s (degree
Degree may refer to:
As a unit of measurement
* Degree symbol (°), a notation used in science, engineering, and mathematics
* Degree (angle), a unit of angle measurement
* Degree (temperature), any of various units of temperature measurement ...

s, revolutions
In political science, a revolution (Latin: ''revolutio'', "a turn around") is a fundamental and relatively sudden change in political power and political organization which occurs when the population revolts against the government, typically due ...

) through which a point revolves around a centre or line has been rotated in a specified sense about a specified axis
Axis may refer to:
Politics
*Axis of evil
The phrase "axis of evil" was first used by U.S. President George W. Bush in his State of the Union address on January 29, 2002, less than five months after the 9/11 attacks, and often repeated t ...

. When a body rotates about its axis, the motion cannot simply be analyzed as a particle, as in circular motion it undergoes a changing velocity and acceleration at any time (''t''). When dealing with the rotation of a body, it becomes simpler to consider the body itself rigid. A body is generally considered rigid when the separations between all the particles remains constant throughout the body's motion, so for example parts of its mass are not flying off. In a realistic sense, all things can be deformable, however this impact is minimal and negligible. Thus the rotation of a rigid body over a fixed axis is referred to as rotational motion
A rotation is a circular movement of an object around a center (or point) of rotation. The geometric plane along which the rotation occurs is called the '' rotation plane'', and the imaginary line extending from the center and perpendicular ...

.
Example

In the example illustrated to the right (or above in some mobile versions), a particle or body P is at a fixed distance ''r'' from the origin, ''O'', rotating counterclockwise. It becomes important to then represent the position of particle P in terms of its polar coordinates (''r'', ''θ''). In this particular example, the value of ''θ'' is changing, while the value of the radius remains the same. (In rectangular coordinates (''x'', ''y'') both ''x'' and ''y'' vary with time). As the particle moves along the circle, it travels anarc length
Arc length is the distance between two points along a section of a curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
In ...

''s'', which becomes related to the angular position through the relationship:-
:$s\; =\; r\backslash theta\; \backslash ,$
Measurements

Angular displacement may be measured inradian
The radian, denoted by the symbol \text, is the SI unit
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

s or degrees. Using radians provides a very simple relationship between distance traveled around the circle and the distance ''r'' from the centre.
:$\backslash theta\; =\; \backslash frac$
For example, if a body rotates 360° around a circle of radius ''r'', the angular displacement is given by the distance traveled around the circumference - which is 2π''r'' - divided by the radius: $\backslash theta=\; \backslash fracr$ which easily simplifies to: $\backslash theta=2\backslash pi$. Therefore, 1 revolution is $2\backslash pi$ radians.
When a particle travels from point P to point Q over $\backslash delta\; t$, as it does in the illustration to the left, the radius of the circle goes through a change in angle $\backslash Delta\; \backslash theta\; =\; \backslash theta\_2\; -\; \backslash theta\_1$ which equals the ''angular displacement''.
Three dimensions

In three dimensions, angular displacement is an entity with a direction and a magnitude. The direction specifies the axis of rotation, which always exists by virtue of theEuler's rotation theorem
In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed poi ...

; the magnitude specifies the rotation in radian
The radian, denoted by the symbol \text, is the SI unit
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

s about that axis (using the right-hand rule
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

to determine direction). This entity is called an axis-angle.
Despite having direction and magnitude, angular displacement is not a vector
Vector may refer to:
Biology
*Vector (epidemiology)
In epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...

because it does not obey the commutative law
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

for addition. Nevertheless, when dealing with infinitesimal rotations, second order infinitesimals can be discarded and in this case commutativity appears.
Several ways to describe angular displacement exist, like rotation matrices or Euler angles
The Euler angles are three angles introduced by Leonhard Euler
Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics ( ...

. See charts on SO(3) for others.
Matrix notation

Given that any frame in the space can be described by a rotation matrix, the displacement among them can also be described by a rotation matrix. Being $A\_0$ and $A\_f$ two matrices, the angular displacement matrix between them can be obtained as $\backslash Delta\; A\; =\; A\_f\; .\; A\_0^$. When this product is performed having a very small difference between both frames we will obtain a matrix close to the identity. In the limit, we will have an infinitesimal rotation matrix.Infinitesimal rotation matrices

An infinitesimal angular displacement is aninfinitesimal rotation In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and t ...

matrix:
* As any rotation matrix has a single real eigenvalue, which is +1, this eigenvalue shows the rotation axis.
* Its module can be deduced from the value of the infinitesimal rotation.
* The shape of the matrix is like this:
:$A\; =\; \backslash begin\; 1\; \&\; -d\backslash phi\_z(t)\; \&\; d\backslash phi\_y(t)\; \backslash \backslash \; d\backslash phi\_z(t)\; \&\; 1\; \&\; -d\backslash phi\_x(t)\; \backslash \backslash \; -d\backslash phi\_y(t)\; \&\; d\backslash phi\_x(t)\; \&\; 1\; \backslash \backslash \; \backslash end$
We can introduce here the infinitesimal angular displacement tensor or rotation generator associated:
:$d\backslash Phi(t)\; =\; \backslash begin\; 0\; \&\; -d\backslash phi\_z(t)\; \&\; d\backslash phi\_y(t)\; \backslash \backslash \; d\backslash phi\_z(t)\; \&\; 0\; \&\; -d\backslash phi\_x(t)\; \backslash \backslash \; -d\backslash phi\_y(t)\; \&\; d\backslash phi\_x(t)\; \&\; 0\; \backslash \backslash \; \backslash end$
Such that its associated rotation matrix is $A\; =\; I\; +\; d\backslash Phi(t)$. When it is divided by the time, this will yield the angular velocity
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

vector.
Generators of rotations

Suppose we specify an axis of rotation by a unit vector 'x'', ''y'', ''z''nbsp;, and suppose we have an infinitely small rotation of angle Δθ about that vector. Expanding the rotation matrix as an infinite addition, and taking the first order approach, the rotation matrix Δ''R'' is represented as: : $\backslash Delta\; R\; =\; \backslash begin\; 1\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \backslash end\; +\; \backslash begin\; 0\; \&\; z\; \&\; -y\; \backslash \backslash \; -z\; \&\; 0\; \&\; x\; \backslash \backslash \; y\; \&\; -x\; \&\; 0\; \backslash end\backslash ,\backslash Delta\; \backslash theta\; =\; \backslash mathbf\; +\; \backslash mathbf\backslash ,\backslash Delta\backslash theta.$ A finite rotation through angle θ about this axis may be seen as a succession of small rotations about the same axis. Approximating Δθ as θ/''N'' where ''N'' is a large number, a rotation of θ about the axis may be represented as: :$R\; =\; \backslash left(\backslash mathbf\; +\; \backslash frac\backslash right)^N\; \backslash approx\; e^.$ It can be seen that Euler's theorem essentially states thatexponential function
The exponential function is a mathematical function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of ...

.
:$\backslash begin\; P\_\; \&=\; -G^2\; \backslash \backslash \; R\; \&=\; I\; -\; P\_\; +\; \backslash left;\; href="/html/ALL/s/I\_\backslash cos\backslash left(\_\backslash beta\_\backslash right)\_+\_G\_\backslash sin\backslash left(\_\backslash beta\_\backslash right)\_\backslash right.html"\; ;"title="I\; \backslash cos\backslash left(\; \backslash beta\; \backslash right)\; +\; G\; \backslash sin\backslash left(\; \backslash beta\; \backslash right)\; \backslash right">I\; \backslash cos\backslash left(\; \backslash beta\; \backslash right)\; +\; G\; \backslash sin\backslash left(\; \backslash beta\; \backslash right)\; \backslash right$
Analysis is often easier in terms of these generators, rather than the full rotation matrix. Analysis in terms of the generators is known as the Lie algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of the rotation group.
Relationship with Lie algebras

The matrices in theLie algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

are not themselves rotations; the skew-symmetric matrices are derivatives, proportional differences of rotations. An actual "differential rotation", or ''infinitesimal rotation matrix'' has the form
:$I\; +\; A\; \backslash ,\; d\backslash theta\; ~,$
where is vanishingly small and , for instance with ,
:$dL\_\; =\; \backslash left;\; href="/html/ALL/s/begin\_1\_\_0\_\_0\_\backslash \backslash \_0\_\_1\_\_-d\backslash theta\_\backslash \backslash \_0\_\_d\backslash theta\_\_1\_\backslash end\backslash right.html"\; ;"title="begin\; 1\; 0\; 0\; \backslash \backslash \; 0\; 1\; -d\backslash theta\; \backslash \backslash \; 0\; d\backslash theta\; 1\; \backslash end\backslash right">begin\; 1\; 0\; 0\; \backslash \backslash \; 0\; 1\; -d\backslash theta\; \backslash \backslash \; 0\; d\backslash theta\; 1\; \backslash end\backslash right$
The computation rules are as usual except that infinitesimals of second order are routinely dropped. With these rules, these matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals. It turns out that ''the order in which infinitesimal rotations are applied is irrelevant''. To see this exemplified, consult infinitesimal rotations SO(3).
Exponential map

Connecting the Lie algebra to the Lie group is the exponential map, which is defined using the standardmatrix exponential
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

series for For any skew-symmetric matrix
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, exp() is always a rotation matrix.Note that this exponential map of skew-symmetric matrices to rotation matrices is quite different from the Cayley transform discussed earlier, differing to 3rd order,
$e^\; -\; \backslash frac=-\; \backslash frac\; A^3\; +\backslash mathrm\; (A^4)\; ~.$ Conversely, a

skew-symmetric matrix
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

specifying a rotation matrix through the Cayley map specifies the ''same'' rotation matrix through the map .
An important practical example is the case. In rotation group SO(3)
In mechanics
Mechanics (Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximatel ...

, it is shown that one can identify every with an Euler vector , where is a unit magnitude vector.
By the properties of the identification , is in the null space of . Thus, is left invariant by and is hence a rotation axis.
Using Rodrigues' rotation formula on matrix form with , together with standard double angle formulae one obtains,
:$\backslash begin\; \backslash exp(\; A\; )\; \&=\; \backslash exp(\backslash theta(\backslash boldsymbol))\; =\; \backslash exp\; \backslash left(\; \backslash left;\; href="/html/ALL/s/begin\_0\_\_-z\_\backslash theta\_\_y\_\backslash theta\_\backslash \backslash \_z\_\backslash theta\_\_0-x\_\backslash theta\_\backslash \backslash \_-y\_\backslash theta\_\_x\_\backslash theta\_\_0\_\backslash end\backslash right.html"\; ;"title="begin\; 0\; -z\; \backslash theta\; y\; \backslash theta\; \backslash \backslash \; z\; \backslash theta\; 0-x\; \backslash theta\; \backslash \backslash \; -y\; \backslash theta\; x\; \backslash theta\; 0\; \backslash end\backslash right">begin\; 0\; -z\; \backslash theta\; y\; \backslash theta\; \backslash \backslash \; z\; \backslash theta\; 0-x\; \backslash theta\; \backslash \backslash \; -y\; \backslash theta\; x\; \backslash theta\; 0\; \backslash end\backslash right$
where , .
This is the matrix for a rotation around axis by the angle in half-angle form. For full detail, see exponential map SO(3).
Notice that for infinitesimal angles second order terms can be ignored and remains exp(A) = I + A
See also

*Angular distance
Angular distance \theta (also known as angular separation, apparent distance, or apparent separation) is the angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a com ...

*Angular position
Changing orientation of a rotating
A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imagina ...

*Angular velocity
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

*Infinitesimal rotation In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and t ...

*Linear elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechan ...

*Second moment of area
The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area
Area is the quantity that expresses the extent of a two-dimensional region, ...

References